Precalculus: pearson new international edition

Michael Sullivan

Chapter 11

Systems of Equations and Inequalities - all with Video Answers

Educators

Chapter Questions

Problem 1

A matrix that has the same number of rows as columns is called a(n) ___________ matrix.

Patrick Burns

Patrick Burns

Numerade Educator

Problem 1

Graph the equation: $y=3 x+2$

Grace Bajar

Numerade Educator

Problem 1

Solve the inequality: $3 x+4<8-x$

Vicki Stebbins

Numerade Educator

Problem 1

A linear programming problem requires that a linear expression, called the , be maximized or minimized.

Rae Xin

Numerade Educator

Problem 1

True or False The equation $(x-1)^2-1=x(x-2)$ is an example of an identity.

Nez Nikoo

Numerade Educator

Problem 1

Solve the equation: $3 x+4=8-x$.

Brandon Fox

Numerade Educator

Problem 1

An $m$ by $n$ rectangular array of numbers is called $a(n)$_______

Christine Anacker

Christine Anacker

Numerade Educator

Problem 1

$D=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=$ ___________

Palitha Reddy

Numerade Educator

Problem 2

True or False Matrix addition is commutative.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 2

Graph the equation: $y+4=x^2$

Grace Bajar

Numerade Educator

Problem 2

Graph the equation: $3 x-2 y=6$

Vicki Stebbins

Numerade Educator

Problem 2

True or False If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points.

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 2

True or False The rational expression $\frac{5 x^2-1}{x^3+1}$ is proper.

QL

Quinn Langford

Numerade Educator

Problem 2

(a) Graph the line: $3 x+4 y=12$.
(b) What is the slope of a line parallel to this line?

Brandon Fox

Numerade Educator

Problem 2

The matrix used to represent a system of linear equations is called a(a)______

Christine Anacker

Christine Anacker

Numerade Educator

Problem 2

Using Cramer's Rule, the value of $x$ that satisfies the system of
$$
\text { equations }\left\{\begin{array}{l}
2 x+3 y=5 \\
x-4 y=-3
\end{array} \text { is } x-\frac{}{\left|\begin{array}{rr}
2 & 3 \\
1 & -4
\end{array}\right|}\right. \text {. }
$$

Palitha Reddy

Numerade Educator

Problem 3

To find the product $A B$ of two matrices $A$ and $B$, the number of __________ in matrix $A$ must equal the number of _________ in matrix $B$.

Michael Jacobsen

Michael Jacobsen

Numerade Educator

Problem 3

Graph the equation: $y^2=x^2-1$

Grace Bajar

Numerade Educator

Problem 3

Graph the equation: $x^2+y^2=9$

Vicki Stebbins

Numerade Educator

Problem 3

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=x+y$

Rae Xin

Numerade Educator

Problem 3

Factor completely: $3 x^4+6 x^3+3 x^2$

QL

Quinn Langford

Numerade Educator

Problem 3

If a system of equations has no solution, it is said to be_____

Brandon Fox

Numerade Educator

Problem 3

The notation ass refers to the entry in the row and column of ______a __________matrix. matrix.

Siena Cizdziel

Numerade Educator

Problem 3

True or False A determinant can never equal 0 .

Sheryl Ezze

Numerade Educator

Problem 4

True or False Matrix multiplication is commutative.

Michael Jacobsen

Michael Jacobsen

Numerade Educator

Problem 4

Graph the equation: $x^2+4 y^2=4$

Grace Bajar

Numerade Educator

Problem 4

Graph the equation: $y=x^2+4$

Vicki Stebbins

Numerade Educator

Problem 4

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=2 x+3 y$

Rae Xin

Numerade Educator

Problem 4

True or False Every polynomial with real numbers as coefficients can be factored into products of linear and/or irreducible quadratic factors.

QL

Quinn Langford

Numerade Educator

Problem 4

If a system of equations has one solution, the system is and the equations are_____

Brandon Fox

Numerade Educator

Problem 4

True or False The matrix $\left[\begin{array}{rr|r}1 & 3 & -2 \\ 0 & 1 & 5 \\ 0 & 0 & 0\end{array}\right]$ is in row

Christine Anacker

Christine Anacker

Numerade Educator

Problem 4

True or False When using Cramer's Rule, if $D=0$, then the system of linear equations is inconsistent.

Palitha Reddy

Numerade Educator

Problem 5

Suppose that $A$ is a square $n$ by $n$ matrix that is nonsingular. The matrix $B$ such that $A B=B A=I_n$ is called the ___________ of the matrix $A$.

Patrick Burns

Numerade Educator

Problem 5

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=x^2+1 \\ y=x+1\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 5

True or False The lines $2 x+y=4$ and $4 x+2 y=0$ are parallel.

Vicki Stebbins

Numerade Educator

Problem 5

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=x+10 y$

Rae Xin

Numerade Educator

Problem 5

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{x}{x^2-1}$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 5

If the solution to a system of two linear equations containing two unknowns is $x=3, y=-2$, then the lines intersect at the point+______

Brandon Fox

Numerade Educator

Problem 5

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{r}x-5 y=5 \\ 4 x+3 y-6\end{array}\right.$

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 5

True or False The value of a determinant remains unchanged if any two rows or any two columns are interchanged.

Palitha Reddy

Numerade Educator

Problem 6

If a matrix $A$ has no inverse, it is called _____________.

Michael Jacobsen

Michael Jacobsen

Numerade Educator

Problem 6

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=x^2+1 \\ y=4 x+1\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 6

The graph of $y=(x-2)^2$ may be obtained by shifting the graph of _____________ to the (left/right) a distance of ____________ units.

Vicki Stebbins

Numerade Educator

Problem 6

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=10 x+y$

Rae Xin

Numerade Educator

Problem 6

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{5 x+2}{x^3-1}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 6

. If the lines that make up a system of two linear equations are coincident, then the system is_____ and ____the equations are

Brandon Fox

Numerade Educator

Problem 6

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{l}3 x+4 y-7 \\ 4 x-2 y=5\end{array}\right.$

Siena Cizdziel

Numerade Educator

Problem 6

True or False If any row (or any column) of a determinant is multiplied by a nonzero number $k$, the value of the determinant remains unchanged.

Palitha Reddy

Numerade Educator

Problem 7

True or False The identity matrix has properties similar to those of the real number 1 .

Patrick Burns

Numerade Educator

Problem 7

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=\sqrt{36-x^2} \\ y=8-x\end{array}\right.$

Check back soon!

Problem 7

When graphing an inequality in two variables, if the inequality is strict use ____________ ; if the inequality is nonstrict use a ___________ mark.

Vicki Stebbins

Numerade Educator

Problem 7

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=5 x+7 y$

Rae Xin

Numerade Educator

Problem 7

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{x^2+5}{x^2-4}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 7

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{array}{l}2 x-y=5 \\ 5 x+2 y=8\end{array}\right.$ $x=2, y=-1 ;(2,-1)$

Brandon Fox

Numerade Educator

Problem 7

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{l}2 x+3 y-6=0 \\ 4 x-6 y+2=0\end{array}\right.$

Jake Zanazzi

Numerade Educator

Problem 7

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rr}6 & 4 \\ -1 & 3\end{array}\right|$

Check back soon!

Problem 8

If $A X=B$ represents a matrix equation where $A$ is a nonsingular matrix, then we can solve the equation using $X=$ __________.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 8

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=\sqrt{4-x^2} \\ y=2 x+4\end{array}\right.$

Check back soon!

Problem 8

The graph of the corresponding equation of a linear inequality is a line that separates the $x y$-plane into two regions. The two regions are called ______________.

Vicki Stebbins

Numerade Educator

Problem 8

In Problems 3-8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points.
(GRAPH CANT COPY)
$z=7 x+5 y$

Rae Xin

Numerade Educator

Problem 8

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{3 x^2-2}{x^2-1}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 8

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 3 x+2 y & =2 \\ x-7 y & =-30\end{aligned}\right.$ $x=-2, y=4 ;(-2,4)$

Brandon Fox

Numerade Educator

Problem 8

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{r}9 x-y=0 \\ 3 x-y-4=0\end{array}\right.$

Lisa Tarman

Numerade Educator

Problem 8

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rr}8 & -3 \\ 4 & 2\end{array}\right|$

Check back soon!

Problem 9

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$
$A+B$

Patrick Burns

Numerade Educator

Problem 9

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=\sqrt{x} \\ y=2-x\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 9

True or False The graph of a system of inequalities must have an overlapping region.

Vicki Stebbins

Numerade Educator

Problem 9

In Problems 9-18, solve each linear programming problem.
Maximize $z=2 x+y$ subject to $x \geq 0, \quad y \geq 0, \quad x+y \leq 6, \quad x+y \geq 1$

Rae Xin

Numerade Educator

Problem 9

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{5 x^3+2 x-1}{x^2-4}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 9

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 3 x-4 y & =4 \\ \frac{1}{2} x-3 y & =-\frac{1}{2}\end{aligned}\right.$
$x=2, y=\frac{1}{2} ;\left(2, \frac{1}{2}\right)$

Brandon Fox

Numerade Educator

Problem 9

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{l}0.01 x-0.03 y=0.06 \\ 0.13 x+0.10 y=0.20\end{array}\right.$

Chelsea Green

Numerade Educator

Problem 9

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rr}-3 & -1 \\ 4 & 2\end{array}\right|$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 10

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$A-B$

Nick Johnson

Numerade Educator

Problem 10

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=\sqrt{x} \\ y=6-x\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 10

If a graph of a system of linear inequalities cannot be contained in any circle, then it is said to be _____________.

Katelyn Vandeaver

Katelyn Vandeaver

Numerade Educator

Problem 10

In Problems 9-18, solve each linear programming problem.
Maximize $z=x+3 y$ subject to $x \geq 0, \quad y \geq 0, x+y \geq 3, \quad x \leq 5, \quad y \leq 7$

Check back soon!

Problem 10

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{3 x^4+x^2-2}{x^3+8}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 10

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 2 x+\frac{1}{2} y & =0 \\ 3 x-4 y & =-\frac{19}{2}\end{aligned}\right.$
$x=-\frac{1}{2}, y=2 ;\left(-\frac{1}{2}, 2\right)$

Brandon Fox

Numerade Educator

Problem 10

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{r}\frac{4}{3} x-\frac{3}{2} y=\frac{3}{4} \\ -\frac{1}{4} x+\frac{1}{3} y=\frac{2}{3}\end{array}\right.$

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 10

In Problems 7-14, find the value of each delerminant.
. $\left|\begin{array}{ll}-4 & 2 \\ -5 & 3\end{array}\right|$

Check back soon!

Problem 11

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$4 A$

Nick Johnson

Numerade Educator

Problem 11

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}x=2 y \\ x=y^2-2 y\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 11

In Problems 11-22, graph each inequality.
$x \geq 0$

Check back soon!

Problem 11

In Problems 9-18, solve each linear programming problem.
Minimize $z=2 x+5 y$ subject to $x \geq 0, \quad y \geq 0, x+y \geq 2, x \leq 5, \quad y \leq 3$

Rae Xin

Numerade Educator

Problem 11

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{x(x-1)}{(x+4)(x-3)}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 11

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{array}{l}x-y=3 \\ \frac{1}{2} x+y=3\end{array}\right.$

Joshua Fischbach

Joshua Fischbach

Numerade Educator

Problem 11

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{r}x-y+z-10 \\ 3 x+3 y=5 \\ x+y+2 z-2\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 11

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rrr}3 & 4 & 2 \\ 1 & -1 & 5 \\ 1 & 2 & -2\end{array}\right

Check back soon!

Problem 12

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$-3 B$

Nick Johnson

Numerade Educator

Problem 12

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=x-1 \\ y=x^2-6 x+9\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 12

In Problems 11-22, graph each inequality.
$y \geq 0$

Check back soon!

Problem 12

In Problems 9-18, solve each linear programming problem.
Minimize $z=3 x+4 y$ subject to $x \geq 0, \quad y \geq 0, \quad 2 x+3 y \geq 6, \quad x+y \leq 8$

Rae Xin

Numerade Educator

Problem 12

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
$\frac{2 x\left(x^2+4\right)}{x^2+1}$

Orlando Nascimento

Orlando Nascimento

Numerade Educator

Problem 12

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{array}{r}x-y=3 \\ -3 x+y=1\end{array}\right.$
$x=4, y=1 ;(4,1)$
$x=-2, y=-5 ;(-2,-5)$

Jake Zanazzi

Numerade Educator

Problem 12

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{r}5 x-y-z=0 \\ x+y=5 \\ 2 x-3 z=2\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 12

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rrr}1 & 3 & -2 \\ 6 & 1 & -5 \\ 8 & 2 & 3\end{array}\right|$

Check back soon!

Problem 13

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$3 A-2 B$

Nick Johnson

Numerade Educator

Problem 13

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{r}x^2+y^2=4 \\ x^2+2 x+y^2=0\end{array}\right.$

Check back soon!

Problem 13

In Problems 11-22, graph each inequality.
$x \geq 4$

Check back soon!

Problem 13

In Problems 9-18, solve each linear programming problem.
Maximize $z=3 x+5 y$ subject to $x \geq 0, \quad y \geq 0, \quad x+y \geq 2, \quad 2 x+3 y \leq 12, \quad 3 x+2 y \leq 12$

Check back soon!

Problem 13

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{4}{x(x-1)}$

Check back soon!

Problem 13

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 3 x+3 y+2 z= & 4 \\ x-y-z= & 0 \\ 2 y-3 z= & -8\end{aligned}\right.$

Brandon Fox

Numerade Educator

Problem 13

In Problems 5-16, write the augmented malrix of the given system of equations
. $\left\{\begin{array}{r}x+y-z-2 \\ 3 x-2 y=2 \\ 5 x+3 y-z=1\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 13

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{lll}4 & -1 & 2 \\ 6 & -1 & 0 \\ 1 & -3 & 4\end{array}\right|$

Check back soon!

Problem 14

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$2 A+4 B$

Nick Johnson

Numerade Educator

Problem 14

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{r}x^2+y^2=8 \\ x^2+y^2+4 y=0\end{array}\right.$

Check back soon!

Problem 14

In Problems 11-22, graph each inequality.
$y \leq 2$

Check back soon!

Problem 14

In Problems 9-18, solve each linear programming problem.
Maximize $z=5 x+3 y$ subject to $x \geq 0, \quad y \geq 0, x+y \geq 2, x+y \leq 8, \quad 2 x+y \leq 10$

Rae Xin

Numerade Educator

Problem 14

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{3 x}{(x+2)(x-1)}$

Check back soon!

Problem 14

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{array}{r}4 x-z=7 \\ 8 x+5 y-z=0 \\ -x-y+5 z=6\end{array}\right.$
$x=1, y=-1, z=2$;
$x=2, y=-3, z=1$;
$(1,-1,2)$
$(2,-3,1)$

Brandon Fox

Numerade Educator

Problem 14

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{rr}2 x+3 y-4 z=0 \\ x-5 z+2=0 \\ x+2 y-3 z=-2\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 14

In Problems 7-14, find the value of each delerminant.
$\left|\begin{array}{rrr}3 & -9 & 4 \\ 1 & 4 & 0 \\ 8 & -3 & 1\end{array}\right|$

Check back soon!

Problem 15

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$A C$

Nick Johnson

Numerade Educator

Problem 15

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} y & =3 x-5 \\ x^2+y^2 & =5\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 15

In Problems 11-22, graph each inequality.
$2 x+y \geq 6$

Check back soon!

Problem 15

In Problems 9-18, solve each linear programming problem.
Minimize $z=5 x+4 y$ subject to $x \geq 0, \quad y \geq 0, \quad x+y \geq 2, \quad 2 x+3 y \leq 12, \quad 3 x+y \leq 12$

Rae Xin

Numerade Educator

Problem 15

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{1}{x\left(x^2+1\right)}$

Check back soon!

Problem 15

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 3 x+3 y+2 z & =4 \\ x-3 y+z & =10 \\ 5 x-2 y-3 z & =8\end{aligned}\right.$

Brandon Fox

Numerade Educator

Problem 15

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{rr}x-y-z & -10 \\ 2 x+y+2 z & =-1 \\ -3 x+4 y & =5\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 15

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}x+y=8 \\ x-y=4\end{array}\right.$

Check back soon!

Problem 16

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$B C$

Nick Johnson

Numerade Educator

Problem 16

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} x^2+y^2 & =10 \\ y & =x+2\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 16

In Problems 11-22, graph each inequality.
$3 x+2 y \leq 6$

Check back soon!

Problem 16

In Problems 9-18, solve each linear programming problem.
Minimize $z=2 x+3 y$ subject to $x \geq 0, \quad y \geq 0, \quad x+y \geq 3, \quad x+y \leq 9, \quad x+3 y \geq 6$

Rae Xin

Numerade Educator

Problem 16

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{1}{(x+1)\left(x^2+4\right)}$

Check back soon!

Problem 16

In Problems 7-16, verify that the values of the variables listed are solutions of the system of equations.
$\left\{\begin{aligned} 4 x-5 z & =6 \\ 5 y-z & =-17 \\ -x-6 y+5 z & =24\end{aligned}\right.$
$x=2, y=-2, z=2 ;(2,-2,2)$
$x=4, y=-3, z=2 ;(4,-3,2)$

Brandon Fox

Numerade Educator

Problem 16

In Problems 5-16, write the augmented malrix of the given system of equations
$\left\{\begin{array}{l}x-y+2 z-w=5 \\ x+3 y-4 z+2 w=2 \\ 3 x-y-5 z-w=-1\end{array}\right.$

Cinsy Krehbiel

Numerade Educator

Problem 16

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}x+2 y-5 \\ x-y=3\end{array}\right.$

Check back soon!

Problem 17

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$C A$

Nick Johnson

Numerade Educator

Problem 17

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{r}x^2+y^2=4 \\ y^2-x=4\end{array}\right.$

Check back soon!

Problem 17

In Problems 11-22, graph each inequality.
$x^2+y^2>1$

Check back soon!

Problem 17

In Problems 9-18, solve each linear programming problem.
Maximize $z=5 x+2 y$ subject to $x \geq 0, \quad y \geq 0, \quad x+y \leq 10,2 x+y \geq 10, \quad x+2 y \geq 10$

Rae Xin

Numerade Educator

Problem 17

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x}{(x-1)(x-2)}$

Check back soon!

Problem 17

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}x+y=8 \\ x-y=4\end{array}\right.$

Hafiz Mohsin Zahid

Hafiz Mohsin Zahid

Numerade Educator

Problem 17

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{ll|r}1 & -3 & -2 \\ 2 & -5 & 5\end{array}\right] \quad R_2--2 r_1+r_2$

Siena Cizdziel

Numerade Educator

Problem 17

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}5 x-y=13 \\ 2 x+3 y=12\end{array}\right.$

Check back soon!

Problem 18

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$C B$

Nick Johnson

Numerade Educator

Problem 18

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}x^2+y^2=16 \\ x^2-2 y=8\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 18

In Problems 11-22, graph each inequality.
$x^2+y^2 \leq 9$

Check back soon!

Problem 18

In Problems 9-18, solve each linear programming problem.
Maximize $z=2 x+4 y$ subject to $x \geq 0, \quad y \geq 0, \quad 2 x+y \geq 4, \quad x+y \leq 9$
864

Rae Xin

Numerade Educator

Problem 18

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{3 x}{(x+2)(x-4)}$

Check back soon!

Problem 18

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
. $\left\{\begin{array}{l}x+2 y=-7 \\ x+y=-3\end{array}\right.$

Check back soon!

Problem 18

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{ll|l}1 & -3 & -3 \\ 2 & -5 & -4\end{array}\right] \quad R_2--2 r_1+r_2$

Siena Cizdziel

Numerade Educator

Problem 18

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+3 y=5 \\ 2 x-3 y=-8\end{array}\right.$

Check back soon!

Problem 19

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$C(A+B)$

Patrick Burns

Numerade Educator

Problem 19

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} x y & =4 \\ x^2+y^2 & =8\end{aligned}\right.$

Check back soon!

Problem 19

In Problems 11-22, graph each inequality.
$y \leq x^2-1$

Check back soon!

Problem 19

Maximizing Profit A manufacturer of skis produces two types: downhill and cross-country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit? What would the maximum profit be if the time available for manufacturing is increased to 48 hours?
$$
\begin{array}{|llll|}
\hline & \text { Downhill } & \begin{array}{c}
\text { Cross- } \\
\text { country }
\end{array} & \begin{array}{c}
\text { Time } \\
\text { Available }
\end{array} \\
\hline \text { Manufacturing time per ski } & 2 \text { hours } & 1 \text { hour } & 40 \text { hours } \\
\text { Finishing time per ski } & 1 \text { hour } & 1 \text { hour } & 32 \text { hours } \\
\text { Profit per ski } & \$ 70 & \$ 50 & \\
\hline
\end{array}
$$

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 19

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2}{(x-1)^2(x+1)}$

Check back soon!

Problem 19

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}5 x-y=21 \\ 2 x+3 y=-12\end{array}\right.$

Check back soon!

Problem 19

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}1 & -3 & 4 & 3 \\ 3 & -5 & 6 & 6 \\ -5 & 3 & 4 & 6\end{array}\right] \begin{aligned} & R_2=-3 r_1+r_2 \\ & R_3-5 r_1+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 19

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}3 x-24 \\ x+2 y=0\end{array}\right.$

Check back soon!

Problem 20

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$(A+B) C$

Patrick Burns

Numerade Educator

Problem 20

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}x^2=y \\ x y=1\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 20

In Problems 11-22, graph each inequality.
$y>x^2+2$

Check back soon!

Problem 20

Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table:
$$
\begin{array}{|lcc|}
\hline & \text { Soybeans } & \text { Wheat } \\
\hline \text { Preparation cost per acre } & \$ 60 & \$ 30 \\
\text { Workdays required per acre } & 3 & 4 \\
\text { Profit per acre } & \$ 180 & \$ 100 \\
\hline
\end{array}
$$
The farmer cannot spend more than $$\$ 1800$$ in preparation costs nor use more than a total of 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit? What is the maximum profit if the farmer is willing to spend no more than $$\$ 2400$$ on preparation?

Rae Xin

Numerade Educator

Problem 20

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x+1}{x^2(x-2)}$

Check back soon!

Problem 20

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{aligned} x+3 y & =5 \\ 2 x-3 y & =-8\end{aligned}\right.$

Check back soon!

Problem 20

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}1 & -3 & 3 & -5 \\ -4 & -5 & -3 & -5 \\ -3 & -2 & 4 & 6\end{array}\right] \begin{aligned} & R_2-4 r_1+r_2 \\ & R_3-3 r_1+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 20

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}4 x+5 y=-3 \\ -2 y=-4\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 21

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$A C-3 I_2$

Nick Johnson

Numerade Educator

Problem 21

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} x^2+y^2 & =4 \\ y & =x^2-9\end{aligned}\right.$

Check back soon!

Problem 21

In Problems 11-22, graph each inequality.
$x y \geq 4$

Check back soon!

Problem 21

Banquet Seating A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $$\$ 28$$ each and 10-person round tables at a cost of $$\$ 52$$ each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The room can have a maximum of 35 tables and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Rae Xin

Numerade Educator

Problem 21

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{1}{x^3-8}$

Check back soon!

Problem 21

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}1 & -3 & 2 & -6 \\ 2 & -5 & 3 & -4 \\ -3 & -6 & 4 & 6\end{array}\right] \begin{aligned} & R_2=-2 r_1+r_2 \\ & R_3-3 r_1+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 21

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}3 x-6 y=24 \\ 5 x+4 y=12\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 22

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$
$\mathrm{CA}+5 I_3$

Nick Johnson

Numerade Educator

Problem 22

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} x y & =1 \\ y & =2 x+1\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 22

In Problems 11-22, graph each inequality.
$x y \leq 1$

Check back soon!

Problem 22

Spring Break The student activities department of a community college plans to rent buses and vans for a springbreak trip. Each bus has 40 regular seats and 1 handicapped seat; each van has 8 regular seats and 3 handicapped seats. The rental cost is $$\$ 350$$ for each van and $$\$ 975$$ for each bus. If 320 regular and 36 handicapped seats are required for the trip, how many vehicles of each type should be rented to minimize cost?

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 22

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{2 x+4}{x^3-1}$

Check back soon!

Problem 22

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}1 & -3 & -4 & -6 \\ 6 & -5 & 6 & -6 \\ -1 & 1 & 4 & 6\end{array}\right] \begin{aligned} & R_2=-6 r_1+r_2 \\ & R_3=r_1+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 22

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}2 x+4 y=16 \\ 3 x-5 y=-9\end{array}\right.$

Check back soon!

Problem 23

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$C A-C B$

Nick Johnson

Numerade Educator

Problem 23

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{array}{l}y=x^2-4 \\ y=6 x-13\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 23

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{r}x+y \leq 2 \\ 2 x+y \geq 4\end{array}\right.$

Check back soon!

Problem 23

Return on Investment An investment broker is instructed by her client to invest up to $$\$ 20,000$$, some in a junk bond yielding $9 \%$ per annum and some in Treasury bills yielding $7 \%$ per annum. The client wants to invest at least $$\$ 8000$$ in T-bills and no more than $$\$ 12,000$$ in the junk bond.
(a) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must equal or exceed the amount placed in junk bonds?
(b) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must not exceed the amount placed in junk bonds?

Rae Xin

Numerade Educator

Problem 23

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2}{(x-1)^2(x+1)^2}$

Check back soon!

Problem 23

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}3 x-6 y=2 \\ 5 x+4 y=1\end{array}\right.$

Check back soon!

Problem 23

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}5 & -3 & 1 & -2 \\ 2 & -5 & 6 & -2 \\ -4 & 1 & 4 & 6\end{array}\right] \begin{aligned} & R_1--2 r_2+r_1 \\ & R_3-2 r_2+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 23

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}3 x-2 y=4 \\ 6 x-4 y-0\end{array}\right.$

Check back soon!

Problem 24

In Problems 9-24, use the following matrices to evaluate the given expression.
$$
A=\left[\begin{array}{rrr}
0 & 3 & -5 \\
1 & 2 & 6
\end{array}\right] \quad B=\left[\begin{array}{rrr}
4 & 1 & 0 \\
-2 & 3 & -2
\end{array}\right] \quad C=\left[\begin{array}{rr}
4 & 1 \\
6 & 2 \\
-2 & 3
\end{array}\right]
$$

$A C+B C$

Patrick Burns

Numerade Educator

Problem 24

In Problems 5-24, graph each equation of the system. Then solve the system to find the points of intersection.
$\left\{\begin{aligned} x^2+y^2 & =10 \\ x y & =3\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 24

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{aligned} 3 x-y & \geq 6 \\ x+2 y & \leq 2\end{aligned}\right.$

Check back soon!

Problem 24

Production Scheduling In a factory, machine 1 produces 8-inch (in.) pliers at the rate of 60 units per hour (hr) and 6 -in. pliers at the rate of $70 \mathrm{units} / \mathrm{hr}$. Machine 2 produces 8 -in. pliers at the rate of 40 units/hr and 6-in. pliers at the rate of 20 units/hr. It costs $$\$ 50 / \mathrm{hr}$$ to operate machine 1 , and machine 2 costs $$\$ 30 / \mathrm{hr}$$ to operate. The production schedule requires that at least 240 units of 8 -in. pliers and at least 140 units of 6-in. pliers be produced during each 10 -hr day. Which combination of machines will cost the least money to operate?

Rae Xin

Numerade Educator

Problem 24

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x+1}{x^2(x-2)^2}$

Check back soon!

Problem 24

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}2 x+4 y=\frac{2}{3} \\ 3 x-5 y=-10\end{array}\right.$

Check back soon!

Problem 24

In Problems 17-24, write the system of equations corresponding to each augmented matrix. Then perform the indicaled row operation(s) on the given augmented matrix.
$\left[\begin{array}{rrr|r}4 & -3 & -1 & 2 \\ 3 & -5 & 2 & 6 \\ -3 & -6 & 4 & 6\end{array}\right] \begin{aligned} & R_1--r_2+r_1 \\ & R_3-r_2+r_3\end{aligned}$

Siena Cizdziel

Numerade Educator

Problem 24

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}-x+2 y=5 \\ 4 x-8 y=6\end{array}\right.$

Check back soon!

Problem 25

In Problems 25-30, find the product.
$\left[\begin{array}{rr}2 & -2 \\ 1 & 0\end{array}\right]\left[\begin{array}{rrrr}2 & 1 & 4 & 6 \\ 3 & -1 & 3 & 2\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 25

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 2 x^2+y^2 & =18 \\ x y & =4\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 25

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}2 x-y \leq 4 \\ 3 x+2 y \geq-6\end{array}\right.$

Check back soon!

Problem 25

Managing a Meat Market A meat market combines ground beef and ground pork in a single package for meat loaf. The ground beef is $75 \%$ lean ( $75 \%$ beef, $25 \%$ fat) and costs the market $$\$ 0.75$$ per pound (lb). The ground pork is $60 \%$ lean and costs the market $$\$ 0.45 / \mathrm{lb}$$. The meat loaf must be at least $70 \%$ lean. If the market wants to use at least $$\$50 \mathrm{lb}$$ of its available pork, but no more than $200 \mathrm{lb}$$ of its available ground beef, how much ground beef should be mixed with ground pork so that the cost is minimized?
(IMAGE CANT COPY)

Sheryl Ezze

Numerade Educator

Problem 25

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x-3}{(x+2)(x+1)^2}$

Check back soon!

Problem 25

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}2 x+y=1 \\ 4 x+2 y=3\end{array}\right.$

Check back soon!

Problem 25

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{ll|r}1 & 0 & 5 \\ 0 & 1 & -1\end{array}\right]$

Check back soon!

Problem 25

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}2 x-4 y=-2 \\ 3 x+2 y=3\end{array}\right.$

Check back soon!

Problem 26

In Problems 25-30, find the product.
$\left[\begin{array}{ll}4 & 1 \\ 2 & 1\end{array}\right]\left[\begin{array}{rrrr}-6 & 6 & 1 & 0 \\ 2 & 5 & 4 & -1\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 26

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}x^2-y^2=21 \\ x+y=7\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 26

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}4 x-5 y \leq 0 \\ 2 x-y \geq 2\end{array}\right.$

Check back soon!

Problem 26

Ice Cream The Mom and Pop Ice Cream Company makes two kinds of chocolate ice cream: regular and premium. The properties of 1 gallon (gal) of each type are shown in the table:
(COLUMN CANT COPY)
In addition, current commitments require the company to make at least $1 \mathrm{gal}$ of premium for every $4 \mathrm{gal}$ of regular. Each day, the company has available 725 pounds (lb) of flavoring and $425 \mathrm{lb}$ of milk-fat products. If the company can ship no more than $3000 \mathrm{lb}$ of product per day, how many gallons of each type should be produced daily to maximize profit?

Rae Xin

Numerade Educator

Problem 26

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2+x}{(x+2)(x-1)^2}$

Check back soon!

Problem 26

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{aligned} x-y & =5 \\ -3 x+3 y & =2\end{aligned}\right.$

Check back soon!

Problem 26

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{ll|r}1 & 0 & -4 \\ 0 & 1 & 0\end{array}\right]$

Check back soon!

Problem 26

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}3 x+3 y=3 \\ 4 x+2 y=\frac{8}{3}\end{array}\right.$

Check back soon!

Problem 27

In Problems 25-30, find the product.
$\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & -1 & 4\end{array}\right]\left[\begin{array}{rr}1 & 2 \\ -1 & 0 \\ 2 & 4\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 27

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} y & =2 x+1 \\ 2 x^2+y^2 & =1\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 27

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}2 x-3 y \leq 0 \\ 3 x+2 y \leq 6\end{array}\right.$

Check back soon!

Problem 27

Maximizing Profit on Ice Skates A factory manufactures two kinds of ice skates: racing skates and figure skates. The racing skates require 6 work-hours in the fabrication department, whereas the figure skates require 4 work-hours there. The racing skates require 1 work-hour in the finishing department, whereas the figure skates require 2 work-hours there. The fabricating department has available at most 120 work-hours per day, and the finishing department has no more than 40 work-hours per day available. If the profit on each racing skate is $$\$ 10$$ and the profit on each figure skate is $$\$ 12$$, how many of each should be manufactured each day to maximize profit? (Assume that all skates made are sold.)

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 27

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x+4}{x^2\left(x^2+4\right)}$

Check back soon!

Problem 27

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent.
. $\left\{\begin{array}{r}2 x-y=0 \\ 4 x+2 y=12\end{array}\right.$

Check back soon!

Problem 27

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{lll|l}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 3\end{array}\right]$

Check back soon!

Problem 27

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}2 x-3 y=-1 \\ 10 x+10 y=5\end{array}\right.$

Check back soon!

Problem 28

In Problems 25-30, find the product.
$\left[\begin{array}{rr}1 & -1 \\ -3 & 2 \\ 0 & 5\end{array}\right]\left[\begin{array}{rrr}2 & 8 & -1 \\ 3 & 6 & 0\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 28

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x^2-4 y^2 & =16 \\ 2 y-x & =2\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 28

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{aligned} 4 x-y & \geq 2 \\ x+2 y & \geq 2\end{aligned}\right.$

Check back soon!

Problem 28

Financial Planning A retired couple has up to $$\$ 50,000$$ to place in fixed-income securities. Their financial adviser suggests two securities to them: one is an AAA bond that yields $8 \%$ per annum; the other is a certificate of deposit $(\mathrm{CD})$ that yields $4 \%$. After careful consideration of the alternatives, the couple decides to place at most $$\$ 20,000$$ in the AAA bond and at least $$\$ 15,000$$ in the $C D$. They also instruct the financial adviser to place at least as much in the $\mathrm{CD}$ as in the AAA bond. How should the financial adviser proceed to maximize the return on their investment?

Rae Xin

Numerade Educator

Problem 28

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{10 x^2+2 x}{(x-1)^2\left(x^2+2\right)}$

Check back soon!

Problem 28

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent
$\left\{\begin{array}{l}3 x+3 y=-1 \\ 4 x+y=\frac{8}{3}\end{array}\right.$

Check back soon!

Problem 28

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{lll|l}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2\end{array}\right]$

Check back soon!

Problem 28

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}3 x-2 y=0 \\ 5 x+10 y=4\end{array}\right.$

Check back soon!

Problem 29

In Problems 25-30, find the product.
$\left[\begin{array}{lll}1 & 0 & 1 \\ 2 & 4 & 1 \\ 3 & 6 & 1\end{array}\right]\left[\begin{array}{rr}1 & 3 \\ 6 & 2 \\ 8 & -1\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 29

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x+y+1 & =0 \\ x^2+y^2+6 y-x & =-5\end{aligned}\right.$

Check back soon!

Problem 29

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{r}x-2 y \leq 6 \\ 2 x-4 y \geq 0\end{array}\right.$

Check back soon!

Problem 29

Product Design An entrepreneur is having a design group produce at least six samples of a new kind of fastener that he wants to market. It costs $$\$ 9.00$$ to produce each metal fastener and $$\$ 4.00$$ to produce each plastic fastener. He wants to have at least two of each version of the fastener and needs to have all the samples 24 hours $(\mathrm{hr})$ from now. It takes $4 \mathrm{hr}$ to produce each metal sample and $2 \mathrm{hr}$ to produce each plastic sample. To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost of the samples?

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 29

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2+2 x+3}{(x+1)\left(x^2+2 x+4\right)}$

Check back soon!

Problem 29

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{rrr|r}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2 \\ 0 & 0 & 0 & 0\end{array}\right]$

Check back soon!

Problem 29

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}2 x+3 y=6 \\ x-y=\frac{1}{2}\end{array}\right.$

Check back soon!

Problem 30

In Problems 25-30, find the product.
$\left[\begin{array}{rrr}4 & -2 & 3 \\ 0 & 1 & 2 \\ -1 & 0 & 1\end{array}\right]\left[\begin{array}{rr}2 & 6 \\ 1 & -1 \\ 0 & 2\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 30

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 2 x^2-x y+y^2 & =8 \\ x y & =4\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 30

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}x+4 y \leq 8 \\ x+4 y \geq 4\end{array}\right.$

Check back soon!

Problem 30

Animal Nutrition Kevin's dog Amadeus likes two kinds of canned dog food. Gourmet Dog costs 40 cents a can and has 20 units of a vitamin complex; the calorie content is 75 calories. Chow Hound costs 32 cents a can and has 35 units of vitamins and 50 calories. Kevin likes Amadeus to have at least 1175 units of vitamins a month and at least 2375 calories during the same time period. Kevin has space to store only 60 cans of dog food at a time. How much of each kind of dog food should Kevin buy each month to minimize his cost?

Ankit Pandey

Numerade Educator

Problem 30

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2-11 x-18}{x\left(x^2+3 x+3\right)}$

Check back soon!

Problem 30

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent
$\left\{\begin{array}{l}3 x-y=7 \\ 9 x-3 y=21\end{array}\right.$

Kelly Hughes

Numerade Educator

Problem 30

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{lll|l}1 & 0 & 4 & 4 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0\end{array}\right]$

Check back soon!

Problem 30

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}\frac{1}{2} x+y=-2 \\ x-2 y=8\end{array}\right.$

Check back soon!

Problem 31

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 31

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 4 x^2-3 x y+9 y^2 & =15 \\ 2 x+3 y & =5\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 31

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}2 x+y \geq-2 \\ 2 x+y \geq 2\end{array}\right.$

Check back soon!

Problem 31

Airline Revenue An airline has two classes of service: first class and coach. Management's experience has been that each aircraft should have at least 8 but no more than 16 first-class seats and at least 80 but not more than 120 coach seats.
(a) If management decides that the ratio of first class to coach seats should never exceed $1: 12$, with how many of each type of seat should an aircraft be configured to maximize revenue?
(b) If management decides that the ratio of first class to coach seats should never exceed 1:8, with how many of each type of seat should an aircraft be configured to maximize revenue?
(c) If you were management, what would you do?

Rae Xin

Numerade Educator

Problem 31

In Problems 13-46, write the partial fraction decomposition of each rational expression.
. $\frac{x}{(3 x-2)(2 x+1)}$

Check back soon!

Problem 31

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent
. $\left\{\begin{aligned} 2 x-3 y & =-1 \\ 10 x+y & =11\end{aligned}\right.$

Check back soon!

Problem 31

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3\end{array}\right]$

Check back soon!

Problem 31

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}3 x-5 y=3 \\ 15 x+5 y=21\end{array}\right.$

Balaji S

Numerade Educator

Problem 32

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rr}3 & -1 \\ -2 & 1\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 32

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{r}2 y^2-3 x y+6 y+2 x+4=0 \\ 2 x-3 y+4=0\end{array}\right.$

Check back soon!

Problem 32

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}x-4 y \leq 4 \\ x-4 y \geq 0\end{array}\right.$

Check back soon!

Problem 32

Explain in your own words what a linear programming problem is and how it can be solved.

Jocelyn Shackelford

Jocelyn Shackelford

Numerade Educator

Problem 32

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{1}{(2 x+3)(4 x-1)}$

Check back soon!

Problem 32

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent
$\left\{\begin{array}{l}3 x-2 y=0 \\ 5 x+10 y=4\end{array}\right.$

Check back soon!

Problem 32

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 & 2 \\ 0 & 0 & 1 & 3 & 0\end{array}\right]$

Check back soon!

Problem 32

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}2 x-y=-1 \\ x+\frac{1}{2} y=\frac{3}{2}\end{array}\right.$

Check back soon!

Problem 33

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{ll}6 & 5 \\ 2 & 2\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 33

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{r}x^2-4 y^2+7=0 \\ 3 x^2+y^2=31\end{array}\right.$

Check back soon!

Problem 33

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}2 x+3 y \geq 6 \\ 2 x+3 y \leq 0\end{array}\right.$

Check back soon!

Problem 33

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x}{x^2+2 x-3}$

Check back soon!

Problem 33

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{llll|l}1 & 0 & 0 & 4 & 2 \\ 0 & 1 & 1 & 3 & 3 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

Check back soon!

Problem 33

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+y-z=6 \\ 3 x-2 y+z=-5 \\ x+3 y-2 z=14\end{array}\right.$

Check back soon!

Problem 34

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rr}-4 & 1 \\ 6 & -2\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 34

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}3 x^2-2 y^2+5=0 \\ 2 x^2-y^2+2=0\end{array}\right.$

Check back soon!

Problem 34

In Problems 23-34, graph each system of linear inequalities.
$\left\{\begin{array}{l}2 x+y \geq 0 \\ 2 x+y \geq 2\end{array}\right.$

Check back soon!

Problem 34

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2-x-8}{(x+1)\left(x^2+5 x+6\right)}$

Check back soon!

Problem 34

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsistent
$\left\{\begin{aligned} \frac{1}{2} x+y & =-2 \\ x-2 y & =8\end{aligned}\right.$

Check back soon!

Problem 34

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 2 & 3\end{array}\right]$

Check back soon!

Problem 34

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{rr}x-y+z=-4 \\ 2 x-3 y+4 z=-15 \\ 5 x+y-2 z=12\end{array}\right.$

Check back soon!

Problem 35

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{ll}2 & 1 \\ a & a\end{array}\right] \quad a \neq 0$

Nick Johnson

Numerade Educator

Problem 35

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 7 x^2-3 y^2+5 & =0 \\ 3 x^2+5 y^2 & =12\end{aligned}\right.$

Check back soon!

Problem 35

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}x^2+y^2 \leq 9 \\ x+y \geq 3\end{array}\right.$

Check back soon!

Problem 35

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2+2 x+3}{\left(x^2+4\right)^2}$

Check back soon!

Problem 35

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}\frac{1}{2} x+\frac{1}{3} y=3 \\ \frac{1}{4} x-\frac{2}{3} y=-1\end{array}\right.$

Check back soon!

Problem 35

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{rrrr|r}1 & 0 & 0 & 1 & -2 \\ 0 & 1 & 0 & 2 & 2 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

Check back soon!

Problem 35

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+2 y-z=-3 \\ 2 x-4 y+z=-7 \\ -2 x+2 y-3 z=4\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 36

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{ll}b & 3 \\ b & 2\end{array}\right] \quad b \neq 0$

Nick Johnson

Numerade Educator

Problem 36

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x^2-3 y^2+1 & =0 \\ 2 x^2-7 y^2+5 & =0\end{aligned}\right.$

Check back soon!

Problem 36

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}x^2+y^2 \geq 9 \\ x+y \leq 3\end{array}\right.$

Check back soon!

Problem 36

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^3+1}{\left(x^2+16\right)^2}$

Check back soon!

Problem 36

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}\frac{1}{3} x-\frac{3}{2} y=-5 \\ \frac{3}{4} x+\frac{1}{3} y=11\end{array}\right.$

Check back soon!

Problem 36

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x, y$; or $x, y$, z; or $x_1, x_2, x_3, x_4$ as variablex Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0\end{array}\right]$

Check back soon!

Problem 36

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+4 y-3 z=-8 \\ 3 x-y+3 z=12 \\ x+y+6 z=1\end{array}\right.$

Check back soon!

Problem 37

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rrr}1 & -1 & 1 \\ 0 & -2 & 1 \\ -2 & -3 & 0\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 37

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}x^2+2 x y=10 \\ 3 x^2-x y=2\end{array}\right.$

Check back soon!

Problem 37

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}y \geq x^2-4 \\ y \leq x-2\end{array}\right.$

Check back soon!

Problem 37

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{7 x+3}{x^3-2 x^2-3 x}$

Check back soon!

Problem 37

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 3 x-5 y & =3 \\ 15 x+5 y & =21\end{aligned}\right.$

Check back soon!

Problem 37

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}x+y=8 \\ x-y=4\end{array}\right.$

Check back soon!

Problem 37

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x-2 y+3 z-1 \\ 3 x+y-2 z-0 \\ 2 x-4 y+6 z-2\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 38

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rrr}1 & 0 & 2 \\ -1 & 2 & 3 \\ 1 & -1 & 0\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 38

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 5 x y+13 y^2+36 & =0 \\ x y+7 y^2 & =6\end{aligned}\right.$

Brittany Scott

Numerade Educator

Problem 38

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}y^2 \leq x \\ y \geq x\end{array}\right.$

Check back soon!

Problem 38

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^3+1}{x^5-x^4}$

Check back soon!

Problem 38

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x-y & =-1 \\ x+\frac{1}{2} y & =\frac{3}{2}\end{aligned}\right.$

Check back soon!

Problem 38

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{rr}x-y+2 z=5 \\ 3 x+2 y=4 \\ -2 x+2 y-4 z=-10\end{array}\right.$

Check back soon!

Problem 39

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rrr}1 & 1 & 1 \\ 3 & 2 & -1 \\ 3 & 1 & 2\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 39

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} 2 x^2+y^2 & =2 \\ x^2-2 y^2+8 & =0\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 39

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}x^2+y^2 \leq 16 \\ y \geq x^2-4\end{array}\right.$

Check back soon!

Problem 39

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2}{x^3-4 x^2+5 x-2}$

Check back soon!

Problem 39

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y}=8 \\ \frac{3}{x}-\frac{5}{y}=0\end{array}\right.$

Check back soon!

Problem 39

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+2 y-z-0 \\ 2 x-4 y+z-0 \\ -2 x+2 y-3 z=0\end{array}\right.$

Yujie Wang

College of San Mateo

Problem 40

In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix.
$\left[\begin{array}{rrr}3 & 3 & 1 \\ 1 & 2 & 1 \\ 2 & -1 & 1\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 40

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} y^2-x^2+4 & =0 \\ 2 x^2+3 y^2 & =6\end{aligned}\right.$

Check back soon!

Problem 40

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}x^2+y^2 \leq 25 \\ y \leq x^2-5\end{array}\right.$

Check back soon!

Problem 40

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2+1}{x^3+x^2-5 x+3}$

Check back soon!

Problem 40

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}\frac{4}{x}-\frac{3}{y}=0 \\ \frac{6}{x}+\frac{3}{2 y}=2\end{array}\right.$

Check back soon!

Problem 40

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x+4 y-3 z=0 \\ 3 x-y+3 z=0 \\ x+y+6 z=0\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 41

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{r}2 x+y=8 \\ x+y=5\end{array}\right.$

Patrick Burns

Numerade Educator

Problem 41

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x^2+2 y^2 & =16 \\ 4 x^2-y^2 & =24\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 41

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{aligned} x y & \geq 4 \\ y & \geq x^2+1\end{aligned}\right.$

Check back soon!

Problem 41

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^3}{\left(x^2+16\right)^3}$

Check back soon!

Problem 41

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x-y & =6 \\ 2 x-3 z & =16 \\ 2 y+z & =4\end{aligned}\right.$

Check back soon!

Problem 41

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x-2 y+3 z=0 \\ 3 x+y-2 z=0 \\ 2 x-4 y+6 z=0\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 42

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{r}3 x-y=8 \\ -2 x+y=4\end{array}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 42

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}4 x^2+3 y^2=4 \\ 2 x^2-6 y^2=-3\end{array}\right.$

Check back soon!

Problem 42

In Problems 35-42, graph each system of inequalities.
$\left\{\begin{array}{l}y+x^2 \leq 1 \\ y \geq x^2-1\end{array}\right.$

Check back soon!

Problem 42

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2}{\left(x^2+4\right)^3}$

Check back soon!

Problem 42

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{r}x-y+2 z=0 \\ 3 x+2 y=0 \\ -2 x+2 y-4 z=0\end{array}\right.$

Check back soon!

Problem 43

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
. $\left\{\begin{aligned} 2 x+y & =0 \\ x+y & =5\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 43

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} \frac{5}{x^2}-\frac{2}{y^2}+3 & =0 \\ \frac{3}{x^2}+\frac{1}{y^2} & =7\end{aligned}\right.$

Check back soon!

Problem 43

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 2 x+y & \leq 6 \\ x+2 y & \leq 6\end{aligned}\right.$

Check back soon!

Problem 43

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{4}{2 x^2-5 x-3}$

Check back soon!

Problem 43

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x-2 y+3 z & =7 \\ 2 x+y+z & =4 \\ -3 x+2 y-2 z & =-10\end{aligned}\right.$

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 43

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}1 & 2 & 3 \\ u & v & w \\ x & y & z\end{array}\right|$

Check back soon!

Problem 44

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{r}3 x-y=4 \\ -2 x+y=5\end{array}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 44

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}\frac{2}{x^2}-\frac{3}{y^2}+1=0 \\ \frac{6}{x^2}-\frac{7}{y^2}+2=0\end{array}\right.$

Check back soon!

Problem 44

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 4 \\ 2 x+3 y & \geq 6\end{aligned}\right.$

Check back soon!

Problem 44

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{4 x}{2 x^2+3 x-2}$

Check back soon!

Problem 44

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x+y-3 z & =0 \\ -2 x+2 y+z & =-7 \\ 3 x-4 y-3 z & =7\end{aligned}\right.$

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 44

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}\frac{1}{2} x+y=-2 \\ x-2 y=8\end{array}\right.$

Check back soon!

Problem 44

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{lll}x & y & z \\ u & v & w \\ 2 & 4 & 6\end{array}\right|$

Check back soon!

Problem 45

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}6 x+5 y=7 \\ 2 x+2 y=2\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 45

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}\frac{1}{x^4}+\frac{6}{y^4}=6 \\ \frac{2}{x^4}-\frac{2}{y^4}=19\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 45

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 2 \\ 2 x+y & \geq 4\end{aligned}\right.$

Check back soon!

Problem 45

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{2 x+3}{x^4-9 x^2}$

Check back soon!

Problem 45

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x-y-z & =1 \\ 2 x+3 y+z & =2 \\ 3 x+2 y & =0\end{aligned}\right.$

Check back soon!

Problem 45

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}3 x-5 y=3 \\ 15 x+5 y=21\end{array}\right.

Check back soon!

Problem 45

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{rrr}x & y & z \\ -3 & -6 & -9 \\ u & v & w\end{array}\right|$

Check back soon!

Problem 46

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned}-4 x+y & =0 \\ 6 x-2 y & =14\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 46

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}\frac{1}{x^4}-\frac{1}{y^4}=1 \\ \frac{1}{x^4}+\frac{1}{y^4}=4\end{array}\right.$

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 46

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+y & \leq 6 \\ 2 x+y & \leq 2\end{aligned}\right.$

Check back soon!

Problem 46

In Problems 13-46, write the partial fraction decomposition of each rational expression.
$\frac{x^2+9}{x^4-2 x^2-8}$

Check back soon!

Problem 46

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x-3 y-z & =0 \\ -x+2 y+z & =5 \\ 3 x-4 y-z & =1\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 46

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}1 & 2 & 3 \\ x-u & y-v & z-w \\ u & v & w\end{array}\right|$

Check back soon!

Problem 47

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}6 x+5 y=13 \\ 2 x+2 y=5\end{array}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 47

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{r}x^2-3 x y+2 y^2=0 \\ x^2+x y=6\end{array}\right.$

Check back soon!

Problem 47

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 2 \\ 2 x+3 y & \leq 12 \\ 3 x+y & \leq
12\end{aligned}\right.$

Check back soon!

Problem 47

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x-y-z= & 1 \\ -x+2 y-3 z= & -4 \\ 3 x-2 y-7 z= & 0\end{aligned}\right.$

Check back soon!

Problem 47

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x-y=6 \\ 2 x-3 z=16 \\ 2 y+z-4\end{array}\right.$

Check back soon!

Problem 47

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}1 & 2 & 3 \\ x-3 & y-6 & z-9 \\ 2 u & 2 v & 2 w\end{array}\right|$

Check back soon!

Problem 48

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned}-4 x+y & =5 \\ 6 x-2 y & =-9\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 48

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x^2-x y-2 y^2 & =0 \\ x y+x+6 & =0\end{aligned}\right.$

Check back soon!

Problem 48

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 2 \\ x+y & \leq 8 \\ 2 x+y & \geq
10\end{aligned}\right.$

Check back soon!

Problem 48

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x-3 y-z & =0 \\ 3 x+2 y+2 z & =2 \\ x+5 y+3 z & =2\end{aligned}\right.$

Jake Zanazzi

Numerade Educator

Problem 48

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{rr}2 x+y= & -4 \\ -2 y+4 z= & 0 \\ 3 x-2 z= & -11\end{array}\right.$

Check back soon!

Problem 48

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}x & y & z-x \\ u & v & w-u \\ 1 & 2 & 2\end{array}\right|$

Check back soon!

Problem 49

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}2 x+y=-3 \\ a x+a y=-a\end{array} \quad a \neq 0\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 49

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} y^2+y+x^2-x-2 & =0 \\ y+1+\frac{x-2}{y} & =0\end{aligned}\right.$

Check back soon!

Problem 49

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 2 \\ x+y & \leq 8 \\ 2 x+y & \leq 10\end{aligned}\right.$

Check back soon!

Problem 49

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x-2 y+3 z & =6 \\ 4 x-3 y+2 z & =0 \\ -2 x+3 y-7 z & =1\end{aligned}\right.$

Victor Salazar

Numerade Educator

Problem 49

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{rr}x-2 y+3 z & =7 \\ 2 x+y+z & =4 \\ -3 x+2 y-2 z & =-10\end{array}\right.$

Check back soon!

Problem 49

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 x & 2 y & 2 z \\ u-1 & v-2 & w-3\end{array}\right|$

Check back soon!

Problem 50

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}b x+3 y=2 b+3 \\ b x+2 y=2 b+2\end{array} \quad b \neq 0\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 50

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} x^3-2 x^2+y^2+3 y-4 & =0 \\ x-2+\frac{y^2-y}{x^2} & =0\end{aligned}\right.$

Check back soon!

Problem 50

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 2 \\ x+y & \leq 8 \\ x+2 y & \geq 1\end{aligned}\right.$

Check back soon!

Problem 50

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}3 x-2 y+2 z=6 \\ 7 x-3 y+2 z=-1 \\ 2 x-3 y+4 z=0\end{array}\right.$

Brittany Scott

Numerade Educator

Problem 50

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}2 x+y-3 z-0 \\ -2 x+2 y+z=-7 \\ 3 x-4 y-3 z=7\end{array}\right.$

Check back soon!

Problem 50

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}x+3 & y+6 & z+9 \\ 3 u-1 & 3 v-2 & 3 w-3 \\ 1 & 2 & 3\end{array}\right|$

Check back soon!

Problem 51

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}2 x+y=\frac{7}{a} \\ a x+a y=5\end{array} \quad a \neq 0\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 51

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} \log _x y & =3 \\ \log _x(4 y) & =5\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 51

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \geq 1 \\ x+2 y & \leq 10\end{aligned}\right.$

Check back soon!

Problem 51

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x+y-z & =6 \\ 3 x-2 y+z & =-5 \\ x+3 y-2 z & =14\end{aligned}\right.$

Victor Salazar

Numerade Educator

Problem 51

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}2 x-2 y-2 z-2 \\ 2 x+3 y+z-2 \\ 3 x+2 y=0\end{array}\right.$

Check back soon!

Problem 51

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{ll}x & x \\ 4 & 3\end{array}\right|-5$

Balaji S

Numerade Educator

Problem 52

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{l}b x+3 y=14 \\ b x+2 y=10\end{array} \quad b \neq 0\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 52

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{array}{l}\log _x(2 y)=3 \\ \log _x(4 y)=2\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 52

In Problems 43-52, graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$\left\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \geq 1 \\ x+2 y & \leq 10 \\ x+y & \geq 2 \\ x+y & \leq 8\end{aligned}\right.$

Check back soon!

Problem 52

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x-y+z & =-4 \\ 2 x-3 y+4 z & =-15 \\ 5 x+y-2 z & =12\end{aligned}\right.$

Check back soon!

Problem 52

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}2 x-3 y-z=0 \\ -x+2 y+z=5 \\ 3 x-4 y-z=1\end{array}\right.$

Check back soon!

Problem 52

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{ll}x & 1 \\ 3 & x\end{array}\right|=-2$

Balaji S

Numerade Educator

Problem 53

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} x-y+z & =0 \\ -2 y+z & =-1 \\ -2 x-3 y & =-5\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 53

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} \ln x & =4 \ln y \\ \log _3 x & =2+2 \log _3 y\end{aligned}\right.$

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 53

In Problems 53-56, write a system of linear inequalities for the given graph.
(GRAPH CANT COPY)

Vicki Stebbins

Numerade Educator

Problem 53

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x+2 y-z & =-3 \\ 2 x-4 y+z & =-7 \\ -2 x+2 y-3 z & =4\end{aligned}\right.$

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 53

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}-x+y+z=-1 \\ -x+2 y=3 z=-4 \\ 3 x-2 y-7 z=0\end{array}\right.$

Check back soon!

Problem 53

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{rrr}x & 1 & 1 \\ 4 & 3 & 2 \\ -1 & 2 & 5\end{array}\right|-2$

Taimoor Shabbir

Taimoor Shabbir

Numerade Educator

Problem 54

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} x+2 z & =6 \\ -x+2 y+3 z & =-5 \\ x-y & =6\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 54

In Problems 25-54, solve each system. Use any method you wish.
$\left\{\begin{aligned} \ln x & =5 \ln y \\ \log _2 x & =3+2 \log _2 y\end{aligned}\right.$

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 54

In Problems 53-56, write a system of linear inequalities for the given graph.
(GRAPH CANT COPY)

Vicki Stebbins

Numerade Educator

Problem 54

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} x+4 y-3 z & =-8 \\ 3 x-y+3 z & =12 \\ x+y+6 z & =1\end{aligned}\right.$

Check back soon!

Problem 54

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}2 x-3 y-z=0 \\ 3 x+2 y+2 z=2 \\ x+5 y+3 z=2\end{array}\right.$

Check back soon!

Problem 54

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{rrr}3 & 2 & 4 \\ 1 & x & 5 \\ 0 & 1 & -2\end{array}\right|=0$

Balaji S

Numerade Educator

Problem 55

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} x-y+z & =2 \\ -2 y+z & =2 \\ -2 x-3 y & =\frac{1}{2}\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 55

Graph the equations given in Example 4.

Jason Taylor-Pestell

Jason Taylor-Pestell

Numerade Educator

Problem 55

In Problems 53-56, write a system of linear inequalities for the given graph.
(GRAPH CANT COPY)

Vicki Stebbins

Numerade Educator

Problem 55

The perimeter of a rectangular floor is 90 feet. Find the dimensions of the floor if the length is twice the width.

Brandon Fox

Numerade Educator

Problem 55

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}2 x-2 y+3 z=6 \\ 4 x-3 y+2 z=0 \\ -2 x+3 y-7 z=1\end{array}\right.$

Check back soon!

Problem 55

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{rrr}x & 2 & 3 \\ 1 & x & 0 \\ 6 & 1 & -2\end{array}\right|=7$

Balaji S

Numerade Educator

Problem 56

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} x+2 z & =2 \\ -x+2 y+3 z & =-\frac{3}{2} \\ x-y & =2\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 56

Graph the equations given in Problem 49.

Jason Taylor-Pestell

Jason Taylor-Pestell

Numerade Educator

Problem 56

In Problems 53-56, write a system of linear inequalities for the given graph.
(GRAPH CANT COPY)

Vicki Stebbins

Numerade Educator

Problem 56

The length of fence required to enclose a rectangular field is 3000 meters What are the dimensions of the field if it is known that the difference between its length and width is 50 meters?

Brandon Fox

Numerade Educator

Problem 56

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}3 x-2 y+2 z-6 \\ 7 x-3 y+2 z=-1 \\ 2 x-3 y+4 z=0\end{array}\right.$

Check back soon!

Problem 56

In Problems 51-56, sotve for $x$.
$\left|\begin{array}{lll}x & 1 & 2 \\ 1 & x & 3 \\ 0 & 1 & 2\end{array}\right|=-4 x$

Balaji S

Numerade Educator

Problem 57

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} x+y+z & =9 \\ 3 x+2 y-z & =8 \\ 3 x+y+2 z & =1\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 57

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{array}{l}y=x^{2 / 3} \\ y=e^{-x}\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 57

Financial Planning A retired couple has up to $$\$ 50,000$$ to invest. As their financial adviser, you recommend that they place at least $$\$ 35,000$$ in Treasury bills yielding $1 \%$ and at most $$\$ 10,000$$ in corporate bonds yielding $3 \%$.
(a) Using $x$ to denote the amount of money invested in Treasury bills and $y$ the amount invested in corporate bonds, write a system of linear inequalities that describes the possible amounts of each investment.
(b) Graph the system and label the corner points.

Vicki Stebbins

Numerade Educator

Problem 57

Orbital Launches In 2005 there was a total of 55 commercial and noncommercial orbital launches worldwide. In addition, the number of noncommercial orbital launches was one more than twice the number of commercial orbital launches. Determine the number of commercial and noncommercial orbital launches in 2005.
Source: Federal Aviation Administration

Kelly Hughes

Numerade Educator

Problem 57

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+y-z=6 \\ 3 x-2 y+z=-5 \\ x+3 y-2 z=14\end{array}\right.$

Check back soon!

Problem 57

Geometry: Equation of a Line An equation of the line containing the two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ may be expressed as the determinant
$$
\left|\begin{array}{lll}
x & y & 1 \\
x_1 & y_1 & 1 \\
x_2 & y_2 & 1
\end{array}\right|=0
$$
Prove this result by expanding the determinant and comparing the result to the two-point form of the equation of a line.

Hoan Nguyen

Numerade Educator

Problem 58

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} 3 x+3 y+z & =8 \\ x+2 y+z & =5 \\ 2 x-y+z & =4\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 58

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{array}{l}y=x^{3 / 2} \\ y=e^{-x}\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 58

Manufacturing Trucks Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours (hr) for painting and $3 \mathrm{hr}$ for detail work; each deluxe model requires $3 \mathrm{hr}$ for painting and $4 \mathrm{hr}$ for detail work. Two painters and three detail workers are employed by the company, and each works $40 \mathrm{hr}$ per week.
(a) Using $x$ to denote the number of standard-model trucks and $y$ to denote the number of deluxe-model trucks, write a system of linear inequalities that describes the possible number of each model of truck that can be manufactured in a week.
(b) Graph the system and label the corner points.

Tawana Stiff

Numerade Educator

Problem 58

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{rr}x-y+z=-4 \\ 2 x-3 y+4 z=-15 \\ 5 x+y-2 z=12\end{array}\right.$

Check back soon!

Problem 58

Geometry: Collinear Points Using the result obtained in Problem 57, show that three distinct points $\left(x_1, y_1\right),\left(x_2, y_2\right)$, and $\left(x_3, y_3\right)$ are collinear (lie on the same line) if and only if
$$
\left|\begin{array}{lll}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|=0
$$

Nick Johnson

Numerade Educator

Problem 59

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{r}x+y+z=2 \\ 3 x+2 y-z=\frac{7}{3} \\ 3 x+y+2 z=\frac{10}{3}\end{array}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 59

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{array}{r}x^2+y^3=2 \\ x^3 y=4\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 59

Blending Coffee Bill's Coffee House, a store that specializes in coffee, has available 75 pounds (lb) of $A$ grade coffee and $120 \mathrm{lb}$ of $B$ grade coffee. These will be blended into 1-lb packages as follows: An economy blend that contains 4 ounces (oz) of $A$ grade coffee and $12 \mathrm{oz}$ of $B$ grade coffee and a superior blend that contains $8 \mathrm{oz}$ of $A$ grade coffee and $8 \mathrm{oz}$ of $B$ grade coffee.
(a) Using $x$ to denote the number of packages of the economy blend and $y$ to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible number of packages of each kind of blend.
(b) Graph the system and label the corner points.

Nick Johnson

Numerade Educator

Problem 59

Movie Theater Tickets A movie theater charges $$\$ 9.00$$ for adults and $$\$ 7.00$$ for senior citizens. On a day when 325 people paid an admission, the total receipts were $$\$2495$$. How many who paid were adults? How many were seniors?

Andrija Isakov

Numerade Educator

Problem 59

Mixing Nuts A store sells cashews for $$\$ 5.00$$ per pound and peanuts for $$\$ 1.50$$ per pound. The manager decides to mix 30 pounds of peanuts with some cashews and sell the mixture for $$\$ 3.00$$ per pound. How many pounds of cashews should be mixed with the peanuts so that the mixture will produce the same revenue as would selling the nuts separately?

Andrija Isakov

Numerade Educator

Problem 59

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+2 y-z=-3 \\ 2 x-4 y+z=-7 \\ -2 x+2 y-3 z=4\end{array}\right.$

Check back soon!

Problem 59

Geometry: Area of a Triangle A triangle has vertices $\left(x_1, y_1\right),\left(x_2, y_2\right)$, and $\left(x_3, y_3\right)$. The area of the triangle is given by the absolute value of $D$, where $D=\frac{1}{2}\left|\begin{array}{ccc}x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1\end{array}\right|$. Use this formula to find the area of a triangle with vertices $(2,3),(5,2)$, and $(6,5)$.

Jessica Delaus

Numerade Educator

Problem 60

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{aligned} 3 x+3 y+z & =1 \\ x+2 y+z & =0 \\ 2 x-y+z & =4\end{aligned}\right.$

Christine Anacker

Christine Anacker

Numerade Educator

Problem 60

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{array}{r}x^3+y^2=2 \\ x^2 y=4\end{array}\right.$

Ziya Ogron

Numerade Educator

Problem 60

Mixed Nuts Nola's Nuts, a store that specializes in selling nuts, has available 90 pounds (lb) of cashews and $120 \mathrm{lb}$ of peanuts. These are to be mixed in 12-ounce (oz) packages as follows: a lower-priced package containing $8 \mathrm{oz}$ of peanuts and $4 \mathrm{oz}$ of cashews and a quality package containing $6 \mathrm{oz}$ of peanuts and $6 \mathrm{oz}$ of cashews.
(a) Use $x$ to denote the number of lower-priced packages and use $y$ to denote the number of quality packages. Write a system of linear inequalities that describes the possible number of each kind of package.
(b) Graph the system and label the corner points.

Tawana Stiff

Numerade Educator

Problem 60

Financial Planning A recently retired couple needs $$\$ 12,000$$ per year to supplement their Social Security. They have $$\$ 150,000$$ to invest to obtain this income. They have decided on two investment options: AA bonds yielding $10 \%$ per annum and a Bank Certificate yielding $5 \%$.
(a) How much should be invested in each to realize exactly $$\$ 12,000$$ ?
(b) If, after 2 years, the couple requires $$\$ 14,000$$ per year in income, how should they reallocate their investment to achieve the new amount?

Brandon Fox

Numerade Educator

Problem 60

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+4 y-3 z=-8 \\ 3 x-y+3 z=12 \\ x+y+6 z=1\end{array}\right.$

Check back soon!

Problem 60

Show that $\left|\begin{array}{lll}x^2 & x & 1 \\ y^2 & y & 1 \\ z^2 & z & 1\end{array}\right|-(y-z)(x-y)(x-z)$.

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 61

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{ll}4 & 2 \\ 2 & 1\end{array}\right]$

Patrick Burns

Numerade Educator

Problem 61

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{aligned} x^4+y^4 & =12 \\ x y^2 & =2\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 61

Transporting Goods A small truck can carry no more than 1600 pounds (lb) of cargo nor more than 150 cubic $\left(\mathrm{ft}^3\right)$ of cargo. A printer weighs $20 \mathrm{lb}$ and occupies $3 \mathrm{ft}^3$ of space. A microwave oven weighs $30 \mathrm{lb}$ and occupies $2 \mathrm{ft}^t$ of space.
(a) Using $x$ to represent the number of microwave ovens and $y$ to represent the number of printers, write a system of linear inequalities that describes the number of ovens and printers that can be hauled by the truck.
(b) Graph the system and label the corner points.

Nick Johnson

Numerade Educator

Problem 61

Computing Wind Speed With a tail wind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the average wind speed and the average airspeed of the Piper.

Erika Bustos

Numerade Educator

Problem 61

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}3 x+y-z-\frac{2}{3} \\ 2 x-y+z-1 \\ 4 x+2 y=\frac{8}{3}\end{array}\right.$

Check back soon!

Problem 61

Complete the proof of Cramer's Rule for two equations containing two variables.
[Hint: In system (5), if $a-0$, then $b \neq 0$ and $c \neq 0$, since $D=-b c \neq 0$. Now show that equation (6) provides a solution of the system when $a-0$. Then three cases remain: $b=0, c-0$, and $d=0$.

Nick Johnson

Numerade Educator

Problem 62

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{rr}-3 & \frac{1}{2} \\ 6 & -1\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 62

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{aligned} x^4+y^4 & =6 \\ x y & =1\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 62

Computing Wind Speed The average airspeed of a singleengine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 62

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+y=1 \\ 2 x-y+z-1 \\ x+2 y+z-\frac{8}{3}\end{array}\right.$

Check back soon!

Problem 62

. Interchange columns 1 and 3 of a 3 by 3 determinant. Show that the value of the new determinant is -1 times the value of the original determinant.

Darian Kaulahao

Darian Kaulahao

Numerade Educator

Problem 63

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{ll}15 & 3 \\ 10 & 2\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 63

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{aligned} x y & =2 \\ y & =\ln x\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 63

Restaurant Management A restaurant manager wants to purchase 200 sets of dishes. One design costs $$\$ 25$$ per set, while another costs $$\$ 45$$ per set. If she only has $\$ 7400$ to spend, how many of each design should be ordered?

Andrija Isakov

Numerade Educator

Problem 63

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{rr}x+y+z+w= & 4 \\ 2 x-y+z= & 0 \\ 3 x+2 y+z-w= & 6 \\ x-2 y-2 z+2 w=-1\end{array}\right.$

Check back soon!

Problem 63

Multiply each entry in row 2 of a 3 by 3 determinant by the number $k, k \neq 0$. Show that the value of the new determinant is $k$ times the value of the original determinant.

Yujie Wang

College of San Mateo

Problem 64

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{rr}-3 & 0 \\ 4 & 0\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 64

In Problems 57-64, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.
$\left\{\begin{aligned} x^2+y^2 & =4 \\ y & =\ln x\end{aligned}\right.$

Ziya Ogron

Numerade Educator

Problem 64

Cost of Fast Food One group of people purchased 10 hot dogs and 5 soft drinks at a cost of $$\$ 35.00$$. A second bought 7 hot dogs and 4 soft drinks at a cost of $$\$ 25.25$$. What is the cost of a single hot dog? A single soft drink?
We paid $$\$ 35.00$$.
How much is one hot dog?
How much is one soda?
We paid $$\$ 25.25$$.
How much is one hot dog?
How much is one soda?

Andrija Isakov

Numerade Educator

Problem 64

Prove that a 3 by 3 determinant in which the entries in column 1 equal those in column 3 has the value 0 .

Darian Kaulahao

Darian Kaulahao

Numerade Educator

Problem 65

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{rrr}-3 & 1 & -1 \\ 1 & -4 & -7 \\ 1 & 2 & 5\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 65

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
The line $x+2 y=0$ and the circle $(x-1)^2+(y-1)^2=5$

Ziya Ogron

Numerade Educator

Problem 65

Computing a Refund The grocery store we use does not mark prices on its goods. My wife went to this store, bought three 1-pound packages of bacon and two cartons of eggs, and paid a total of $$\$ 13.45$$. Not knowing that she went to the store, I also went to the same store, purchased two 1-pound packages of bacon and three cartons of eggs, and paid a total of $$\$ 11.45$$. Now we want to return two 1-pound packages of bacon and two cartons of eggs. How much will be refunded?

Erika Bustos

Numerade Educator

Problem 65

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+2 y+z-1 \\ 2 x-y+2 z-2 \\ 3 x+y+3 z=3\end{array}\right.$

Check back soon!

Problem 65

Prove that, if row 2 of a 3 by 3 determinant is multiplied by $k, k \neq 0$, and the result is added to the entries in row 1 , there is no change in the value of the determinant.

Yujie Wang

College of San Mateo

Problem 66

In Problems 61-66, show that each matrix has no inverse.
$\left[\begin{array}{rrr}1 & 1 & -3 \\ 2 & -4 & 1 \\ -5 & 7 & 1\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 66

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
The line $x+2 y+6=0$ and the circle $(x+1)^2+(y+1)^2=5$

Ziya Ogron

Numerade Educator

Problem 66

Finding the Current of a Stream Pamela requires 3 hours to swim 15 miles downstream on the Illinois River. The return trip upstream takes 5 hours. Find Pamela's average speed in still water. How fast is the current? (Assume that Pamela's speed is the same in each direction.)

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 66

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+2 y-z=3 \\ 2 x-y+2 z=6 \\ x-3 y+3 z=4\end{array}\right.$

Check back soon!

Problem 67

In Problems 67-70, use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.
$\left[\begin{array}{rrr}25 & 61 & -12 \\ 18 & -2 & 4 \\ 8 & 35 & 21\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 67

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
The circle $(x-1)^2+(y+2)^2=4$ and

Ziya Ogron

Numerade Educator

Problem 67

Pharmacy A doctor's prescription calls for a daily intake containing 40 milligrams ( $\mathrm{mg}$ ) of vitamin $\mathrm{C}$ and $30 \mathrm{mg}$ of vitamin D. Your pharmacy stocks two liquids that can be used: one contains $20 \%$ vitamin $C$ and $30 \%$ vitamin $D$, the other $40 \%$ vitamin C and $20 \%$ vitamin D. How many milligrams of each compound should be mixed to fill the prescription?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 67

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x-y+z-5 \\ 3 x+2 y-2 z=0\end{array}\right.$

Check back soon!

Problem 68

In Problems 67-70, use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.
$\left[\begin{array}{rrr}18 & -3 & 4 \\ 6 & -20 & 14 \\ 10 & 25 & -15\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 68

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
The circle $(x+2)^2+(y-1)^2=4$ and the parabola $y^2+4 y-x+1=0$ the parabola $y^2-2 y-x-5=0$

Ziya Ogron

Numerade Educator

Problem 68

Pharmacy A doctor's prescription calls for the creation of pills that contain 12 units of vitamin $B_{12}$ and 12 units of vitamin E. Your pharmacy stocks two powders that can be used to make these pill: one contains $20 \%$ vitamin $\mathrm{B}_p$ and $30 \%$ vitamin $\mathrm{E}$, the other $40 \%$ vitamin $\mathrm{B}_{12}$ and $20 \%$ vitamin E. How many units of each powder should be mixed in each pill?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 68

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}2 x+y-z-4 \\ -x+y+3 z-1\end{array}\right.$

Check back soon!

Problem 69

In Problems 67-70, use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.
$\left[\begin{array}{rrrr}44 & 21 & 18 & 6 \\ -2 & 10 & 15 & 5 \\ 21 & 12 & -12 & 4 \\ -8 & -16 & 4 &
9\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 69

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
$y=\frac{4}{x-3}$ and the circle $x^2-6 x+y^2+1=0$

Ziya Ogron

Numerade Educator

Problem 69

Curve Fitting Find real numbers $a, b$, and $c$ so that the graph of the function $y=a x^2+b x+c$ contains the points $(-1,4),(2,3)$, and $(0,1)$.

Erika Bustos

Numerade Educator

Problem 69

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}2 x+3 y-z-3 \\ x-y-z=0 \\ -x+y+z-0 \\ x+y+3 z-5\end{array}\right.$

Check back soon!

Problem 70

In Problems 67-70, use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.
$\left[\begin{array}{rrrr}16 & 22 & -3 & 5 \\ 21 & -17 & 4 & 8 \\ 2 & 8 & 27 & 20 \\ 5 & 15 & -3 & -10\end{array}\right]$

Kyle Christian

Numerade Educator

Problem 70

In Problems 65-70, graph each equation and find the point(s) of intersection, if any.
$y=\frac{4}{x+2}$ and the circle $x^2+4 x+y^2-4=0$

Ziya Ogron

Numerade Educator

Problem 70

Curve Fitting Find real numbers $a, b$, and $c$ so that the graph of the function $y=a x^2+b x+c$ contains the points $(-1,-2),(1,-4)$, and $(2,4)$.

Erika Bustos

Numerade Educator

Problem 70

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x-3 y+z=1 \\ 2 x-y-4 z=0 \\ x-3 y+2 z=1 \\ x-2 y=5\end{array}\right.$

Check back soon!

Problem 71

In Problems 71-74, use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\left\{\begin{aligned} 25 x+61 y-12 z & =10 \\ 18 x-12 y+7 y & =-9 \\ 3 x+4 y-z & =12\end{aligned}\right.$

Patrick Burns

Numerade Educator

Problem 71

The difference of two numbers is 2 and the sum of their squares is 10 . Find the numbers.

Brittany Scott

Numerade Educator

Problem 71

IS-LM Model in Eeonomics In economica, the IS curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for goods in the economy. The L.M curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilihrium level of income (in millions of dollars) and interest rates satisfy the system of equations
$$
\left\{\begin{array}{l}
0.06 Y-5000 r=240 \\
0.06 Y+6000 r=900
\end{array}\right.
$$
Find the equilibrium level of income and interest rates.

Erika Bustos

Numerade Educator

Problem 71

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}4 x+y+z-w=4 \\ x-y+2 z+3 w=3\end{array}\right.$

Check back soon!

Problem 72

In Problems 71-74, use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\left\{\begin{aligned} 25 x+61 y-12 z & =15 \\ 18 x-12 y+7 z & =-3 \\ 3 x+4 y-z & =12\end{aligned}\right.$

Melissa Barry

Numerade Educator

Problem 72

The sum of two numbers is 7 and the difference of their squares is 21 . Find the numbers.

Brittany Scott

Numerade Educator

Problem 72

IS-LM Model in Economics In economics, the IS curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for goods in the economy. The L.M curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilihrium level of income (in millions of dollars) and interest rates satisfy the system of equations
$$
\left\{\begin{array}{l}
0.05 Y-1000 r=10 \\
0.05 Y+800 r=100
\end{array}\right.
$$
Find the equilibrium level of income and interest rates.

Erika Bustos

Numerade Educator

Problem 72

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}-4 x+y=5 \\ 2 x-y+z-w=5 \\ z+w=4\end{array}\right.$

Check back soon!

Problem 73

In Problems 71-74, use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\left\{\begin{aligned} 25 x+61 y-12 z & =21 \\ 18 x-12 y+7 z & =7 \\ 3 x+4 y-z & =-2\end{aligned}\right.$

Melissa Barry

Numerade Educator

Problem 73

The product of two numbers is 4 and the sum of their squares is 8 . Find the numbers.

Brittany Scott

Numerade Educator

Problem 73

Electricity: Kirchhoff's Rules An application of Kirchhoffs Rules to the circuit shown on the next page results in the following system of equations:
$$
\left\{\begin{aligned}
I_2 & =I_1+I_3 \\
5-3 I_1-5 I_2 & =0 \\
10-5 I_2-7 I_3 & =0
\end{aligned}\right.
$$
Find the currents $I_1, I_2$, and $I_3$.

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 73

Curve Fitting Find the function $y-a x^2+b x+c$ whose graph contains the points $(1,2),(-2,-7)$, and $(2,-3)$.

Christine Anacker

Christine Anacker

Numerade Educator

Problem 74

In Problems 71-74, use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\left\{\begin{aligned} 25 x+61 y-12 z & =25 \\ 18 x-12 y+7 z & =10 \\ 3 x+4 y-z & =-4\end{aligned}\right.$

Melissa Barry

Numerade Educator

Problem 74

The product of two numbers is 10 and the difference of their squares is 21 . Find the numbers.

Madi Sousa

Numerade Educator

Problem 74

Electricityt Kirchhoff's Rules An application of Kirchhoffs Rules to the circuit shown results in the following system of equations:
$$
\left\{\begin{array}{c}
I_3=I_1+I_2 \\
8=4 I_3+6 I_2 \\
8 I_1=4+6 I_2
\end{array}\right.
$$
Find the currents $I_1, I_2$, and $I_3$.
Sounce Physics for Scientists \& Engineers, 3rd ed., by Serway. D 1990 Brooks/Cole, a division of Thomson Learning.

Andrija Isakov

Numerade Educator

Problem 74

Curve Fitting Find the function $y-a x^2+b x+c$ whose graph contains the points $(1,-1),(3,-1)$, and $(-2,14)$.

Erika Bustos

Numerade Educator

Problem 75

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{array}{l}2 x+3 y=11 \\ 5 x+7 y=24\end{array}\right.$

Nick Johnson

Numerade Educator

Problem 75

The difference of two numbers is the same as their product, and the sum of their reciprocals is 5 . Find the numbers.

Brittany Scott

Numerade Educator

Problem 75

Theater Revenues A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $$\$ 50$$, main seats for $$\$ 35$$, and balcony seats for $$\$25$$. If all the seats are sold, the gross revenue to the theater is $$\$ 17,100$$. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $$\$ 14,600$$. How many are there of each kind of seat?

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 75

Curve Fitting Find the function $f(x)-a x^3+b x^2+$ $c x+d$ for which $f(-3)--112, f(-1)=-2, f(1)-4$, and $f(2)-13$.

Check back soon!

Problem 76

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned} 2 x+8 y & =-8 \\ x+7 y & =-13\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 76

The sum of two numbers is the same as their product, and the difference of their reciprocals is 3 . Find the numbers.

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 76

Theater Revenues A movie theater charges $$\$ 8.00$$ for adults, $$\$ 4.50$$ for children, and $$\$ 6.00$$ for senior citizens. One day the theater sold 405 tickets and collected $$\$ 2320$$ in receipts. Twice as many children's tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 76

Curve Fitting Find the function $f(x)-a x^3+b x^2+$ $c x+d$ for which $f(-2)=-10, f(-1)-3, f(1)-5$, and $f(3)=15$.

Ashly Sunny

Numerade Educator

Problem 77

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned} x-2 y+4 z & =2 \\ -3 x+5 y-2 z & =17 \\ 4 x-3 y & =-22\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 77

The ratio of $a$ to $b$ is $\frac{2}{3}$. The sum of $a$ and $b$ is 10 . What is the ratio of $a+b$ to $b-a$ ?

Brittany Scott

Numerade Educator

Problem 77

Nutrition $A$ dietitian wishes a patient to have a meal that has 66 grams ( $\mathrm{g}$ ) of protein, $94.5 \mathrm{~g}$ of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, eorn, and $2 \%$ milk. Each serving of chicken has $30 \mathrm{~g}$ of protein, $35 \mathrm{~g}$ of carbohydrates, and $200 \mathrm{mg}$ of calcium. Each serving of corn has $3 \mathrm{~g}$ of protein, $16 \mathrm{~g}$ of carbohydrates, and $10 \mathrm{mg}$ of calcium. Each glass of $2 \%$ milk has $9 \mathrm{~g}$ of protein, $13 \mathrm{~g}$ of carbohydrates, and $300 \mathrm{mg}$ of calcium. How many servings of each food should the dietitian provide for the patient?

David Mccaslin

Numerade Educator

Problem 77

Netrition A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams $(\mathrm{g})$ of protein, $59 \mathrm{~g}$ of carbohydrates, and 75 milligrams ( $\mathrm{mg}$ ) of vitamin $\mathrm{A}$. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has $30 \mathrm{~g}$ of protein, $20 \mathrm{~g}$ of carbohydrates, and $2 \mathrm{mg}$ of vitamin A. Each serving of baked eggs contains $15 \mathrm{~g}$ of protein, $2 \mathrm{~g}$ of carbohydrates, and $20 \mathrm{mg}$ of vitamin A. Each serving of acorn squash contains $3 \mathrm{~g}$ of protein, $25 \mathrm{~g}$ of carbohydrates, and $32 \mathrm{mg}$ of vitamin $\mathrm{A}$. How many servings of each food should the dietitian provide for the patient?

Rd

Reinhart Du Plessis

Numerade Educator

Problem 78

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned} 2 x+3 y-z & =-2 \\ 4 x+3 z & =6 \\ 6 y-2 z & =2\end{aligned}\right.$

Nick Johnson

Numerade Educator

Problem 78

The ratio of $a$ to $b$ is $4: 3$. The sum of $a$ and $b$ is 14 . What is the ratio of $a-b$ to $a+b$ ?

Brittany Scott

Numerade Educator

Problem 78

Investments Kelly has $$\$ 20,000$$ to invest. As her financial planner, you recommend that she diversify into three invest. ments. Treasury bills that yield $5 \%$ simple interest, Treasury bonds that yield $7 \%$ simple interest, and corporate bonds that yield $10 \%$ simple interest. Kelly wishes to earn $$\$ 1390$$ per year in income. Also, Kelly wants her investment in Treasury bills to be $$\$ 3000$$ more than her investment in corporate bonds. How much money should Kelly place in each investment?

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 78

Nutrition A dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, $58 \mathrm{~g}$
of carbohydrates, and 630 milligrams ( $\mathrm{mg}$ ) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and $2 \%$ milk. Each serving of pork chops has $23 \mathrm{~g}$ of protein, $0 \mathrm{~g}$ of carbohydrates, and $10 \mathrm{mg}$ of calcium. Each serving of com on the cob contains $3 \mathrm{~g}$ of protein, $16 \mathrm{~g}$ of carbohydrates, and $10 \mathrm{mg}$ of calcium. Each glass of $2 \%$ milk contains $9 \mathrm{~g}$ of protein, $13 \mathrm{~g}$ of carbohydrates, and $300 \mathrm{mg}$ of calcium. How many servings of each food should the dietitian provide for the patient?

Christine Anacker

Christine Anacker

Numerade Educator

Problem 79

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned} 5 x-y+4 z & =2 \\ -x+5 y-4 z & =3 \\ 7 x+13 y-4 z & =17\end{aligned}\right.$

Melvin Adkins

Numerade Educator

Problem 79

Geometry The perimeter of a rectangle is 16 inches and its area is 15 square inches. What are its dimensions?

Brittany Scott

Numerade Educator

Problem 79

Prices of Fust Food One group of customers bought 8 deluxe hamburgers, 6 orders of large fries, and 6 large colas for $$\$26.10.$$ A second group ordered 10 deluxe hamburgers, 6 large fries, and 8 large colas and paid $$\$ 31.60$$. Is there sufficient information to determine the price of each food item? If not, construct a table showing the various possibilities. Assume that the hamburgers cost between $$\$ 1.75$$ and $$\$ 2.25$$, the fries between $$$\$ 0.75$$ and $$\$ 1.00$$, and the colas between $$\$ 0.60$$ and $$\$ 0.90$$.

Darssan Eswaramoorthi

Darssan Eswaramoorthi

River Bluff High School

Problem 79

Financial Planning Carletta has $\$ 10,000$ to invest. As ber financial consultant, you recommend that she invest in Treasury bills that yield $6 \%$, Treasury bonds that yield $7 \%$, and corporate bonds that yield $8 \%$. Carletta wants to have an annual income of $\$ 680$, and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount in each investment.

Rd

Reinhart Du Plessis

Numerade Educator

Problem 80

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{array}{l}3 x+2 y-z=2 \\ 2 x+y+6 z=-7 \\ 2 x+2 y-14 z=17\end{array}\right.$

Melvin Adkins

Numerade Educator

Problem 80

Geometry An area of 52 square feet is to be enclosed by two squares whose sides are in the ratio of $2: 3$. Find the sides of the squares.

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 80

Prices of Fust Food Use the information given in Problem 79. Suppose that a third group purchased 3 deluxe hamburgers, 2 large fries, and 4 large colas for $$\$10.95$$. Now is there sufficient information to determine the price of each food item? If $\mathrm{sa}$, determine each price.
81. Painting a House Three painters, Beth, Bell, and Edie, working together, can paint the exterior of a home in 10 hours (hr). Bill and Edie together have painted a similar house in $15 \mathrm{hr}$. One day, all three worked on this same kind of house for $4 \mathrm{hr}$, after which Edie left. Beth and Bill required 8 more hr to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job alone?

Darssan Eswaramoorthi

Darssan Eswaramoorthi

River Bluff High School

Problem 80

Landscaping A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of $$\$ 7520$$. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of $$\$ 5945$$. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $$\$ 8985$$. Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge. Sources: www.bx.org

Oluwadamilola Ameobi

Oluwadamilola Ameobi

Numerade Educator

Problem 81

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned} 2 x-3 y+z & =4 \\ -3 x+2 y-z & =-3 \\ -5 y+z & =6\end{aligned}\right.$

Melvin Adkins

Numerade Educator

Problem 81

Geometry Two circles have circumferences that add up to $12 \pi$ centimeters and areas that add up to $20 \pi$ square centimeters. Find the radius of each circle.

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 81

Painting a House Three painters, Beth, Bell, and Edie, working together, can paint the exterior of a home in 10 hours (hr). Bill and Edie together have painted a similar house in $15 \mathrm{hr}$. One day, all three worked on this same kind of house for $4 \mathrm{hr}$, after which Edie left. Beth and Bill required 8 more hr to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job alone?

Jessica Bunge

Numerade Educator

Problem 81

Production To manufacture an automobile requires painting. drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (hr) for painting, $3 \mathrm{hr}$ for drying, and $2 \mathrm{hr}$ for polishing. A Beta requires 16 hr for painting, 5 hr for drying, and $3 \mathrm{hr}$ for polishing, and a Sigma requires $8 \mathrm{hr}$ for painting, 2 hr for drying, and 1 hr for polishing. If the company has $240 \mathrm{hr}$ for painting, $69 \mathrm{hr}$ for drying, and $41 \mathrm{hr}$ for polishing per month, how many of each type of car are produced?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 82

In Problems $75-82$, algebraically solve each system of equations using any method you wish.
$\left\{\begin{aligned}-4 x+3 y+2 z & =6 \\ 3 x+y-z & =-2 \\ x+9 y+z & =6\end{aligned}\right.$

Melvin Adkins

Numerade Educator

Problem 82

Geometry The altitude of an isosceles triangle drawn to its base is 3 centimeters, and its perimeter is 18 centimeters. Find the length of its base.

Sushmit Acharya

Sushmit Acharya

Numerade Educator

Problem 82

. Make up a system of three linear equations containing three variables that has:
(a) No solution
(b) Exactly one solution
(c) Infinitely many solutions
Give the three systems to a friend to solve and critique.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 82

Production A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, $6 \mathrm{~min}$ for filling, and $1 \mathrm{~min}$ for labeling. Each case of grapefruit juice requires $10 \mathrm{~min}$ for sterilizing, 4 min for filling, and $2 \mathrm{~min}$ for labeling. Each case of tomato juice requires $12 \mathrm{~min}$ for sterilizing, $4 \mathrm{~min}$ for filling, and $1 \mathrm{~min}$ for labeling. If the company runs the sterilizing machine for $398 \mathrm{~min}$, the filling machine for $164 \mathrm{~min}$, and the labeling machine for 58 min, how many cases of each type of juice are prepared?

Christine Anacker

Christine Anacker

Numerade Educator

Problem 83

College Tuition Nikki and Joe take classes at a community college, LCCC, and a local university, SIUE. The number of credit hours taken and the cost per credit hour (2009-2010 academic year, tuition only) are as follows:
$$
\begin{array}{|lcc|}
\hline & \text { LCCC } & \text { SIUE } \\
\hline \text { Nikki } & 6 & 9 \\
\text { Joe } & 3 & 12 \\
\hline
\end{array}
$$

$$
\begin{array}{|lc|}
\hline & \text { Cost per } \\
\hline \text { Credit Hour } \\
\hline \text { LCCC } & \$ 80.00 \\
\text { SIUE } & \$ 277.80 \\
\hline
\end{array}
$$
(a) Write a matrix $A$ for the credit hours taken by each student and a matrix $B$ for the cost per credit hour.
(b) Compute $A B$ and interpret the results.

Kelly Korek

Numerade Educator

Problem 83

The Tortoise and the Hare In a 21-meter race between a tortoise and a hare, the tortoise leaves 9 minutes before the hare. The hare, by running at an average speed of 0.5 meter per hour faster than the tortoise, crosses the finish line 3 minutes before the tortoise. What are the average speeds of the tortoise and the hare?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 83

. Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 83

Electricity: Kirchhofr's Rules An application of Kirchhotf's Rules to the circuit shown results in the following system of equations:
$$
\left\{\begin{aligned}
-4+8-2 I_2 & =0 \\
8 & =5 I_4+I_1 \\
4 & =3 I_3+I_1 \\
I_3+I_4 & =I_1
\end{aligned}\right.
$$
Find the currents $I_1, I_2, I_3$, and $I_4$.
Source: Based on Raymond Serway. Physics, 3rd ed. (Philadelphia: Saunders, 1990), Prob. 34. p. 790.

David Mccaslin

Numerade Educator

Problem 84

School Loan Interest Jamal and Stephanie each have school loans issued from the same two banks. The amounts borrowed and the monthly interest rates are given next (interest is compounded monthly):
$$
\begin{array}{|lcc|}
\hline & \text { Lender } \mathbf{1} & \text { Lender } \mathbf{2} \\
\hline \text { Jamal } & \$ 4000 & \$ 3000 \\
\text { Stephanie } & \$ 2500 & \$ 3800 \\
\hline
\end{array}
$$
$$
\begin{array}{|lc|}
\hline & \text { Monthly Interest Rate } \\
\hline \text { Lender } 1 & 0.011(1.1 \%) \\
\text { Lender } 2 & 0.006(0.6 \%) \\
\hline
\end{array}
$$
(a) Write a matrix $A$ for the amounts borrowed by each student and a matrix $B$ for the monthly interest rates.
(b) Compute $A B$ and interpret the results.
(c) Let $C=\left[\begin{array}{l}1 \\ 1\end{array}\right]$. Compute $A(C+B)$ and interpret the results.

Emily Quinn

Numerade Educator

Problem 84

Running a Race In a 1-mile race, the winner crosses the finish line 10 feet ahead of the second-place runner and 20 feet ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how many feet does the second-place runner beat the third-place runner?

Ziya Ogron

Numerade Educator

Problem 84

Do you prefer the method of substitution or the method of elimination for solving a system of two linear equations containing two variables? Give reasons.

Brandon Fox

Numerade Educator

Problem 84

Electricity: Kirchhofr's Rules An application of Kirchhoff's Rules to the circuit shown results in the following system of equations:
$$
\left\{\begin{aligned}
I_1 & =I_3+I_2 \\
24-6 I_1-3 I_3 & =0 \\
12+24-6 I_1-6 I_2 & =0
\end{aligned}\right.
$$
Find the currents $I_1 I_2$, and $I_2$.

David Mccaslin

Numerade Educator

Problem 85

Computing the Cost of Production The Acme Steel Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10-gallon (gal) capacity, 350 with 5 -gal capacity, and 400 with 1 -gal capacity. On the same day, the following aluminum containers were manufactured: 700 with 10 -gal capacity, 500 with 5 -gal capacity, and 850 with 1 -gal capacity.
(a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data.
(b) If the amount of material used in the 10-gal containers is 15 pounds ( $\mathrm{lb}$ ), the amount used in the 5-gal containers is $8 \mathrm{lb}$, and the amount used in the 1 -gal containers is $3 \mathrm{lb}$, find a 3 by 1 matrix representing the amount of material used.
(c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day's usage of material.
(d) If stainless steel costs Acme $$\$ 0.10 \mathrm{lb}$$ and aluminum costs $$\$ 0.05 \mathrm{lb}$$, find a 1 by 2 matrix representing cost.
(e) Multiply the matrices found in parts (c) and (d) to determine the total cost of the day's production.

David Mccaslin

Numerade Educator

Problem 85

Constructing a Box A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2-centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Jason Taylor-Pestell

Jason Taylor-Pestell

Numerade Educator

Problem 85

Financial Planning Three retired couples each require an additional annual income of $\$ 2000$ per year. As their financial consultant, you recommend that they invest some money in Treasury bills that yield $7 \%$, some money in corporate bonds that yield $9 \%$, and some money in junk bonds that yield $11 \%$. Prepare a table for each couple showing the various ways that their goals can be achieved:
(a) If the first couple has $\$ 20,000$ to invest.
(b) If the second couple has $\$ 25,000$ to invest.
(c) If the third couple has $\$ 30,000$ to invest.
(d) What advice would you give each couple regarding the amount to invest and the choices available?
[Hint: Higher yields generally carry more risk.]

Christine Anacker

Christine Anacker

Numerade Educator

Problem 86

Computing Profit Rizza Ford has two locations, one in the city and the other in the suburbs. In January, the city location sold 400 subcompacts, 250 intermediate-size cars, and 50 SUVs; in February, it sold 350 subcompacts, 100 intermediates, and 30 SUVs. At the suburban location in January, 450 subcompacts, 200 intermediates, and 140 SUVs were sold. In February, the suburban location sold 350 subcompacts, 300 intermediates, and 100 SUVs.
(a) Find 2 by 3 matrices that summarize the sales data for each location for January and February (one matrix for each month).
(b) Use matrix addition to obtain total sales for the 2-month period.
(c) The profit on each kind of car is $\$ 100$ per subcompact, $\$ 150$ per intermediate, and $\$ 200$ per SUV. Find a 3 by 1 matrix representing this profit.
(d) Multiply the matrices found in parts (b) and (c) to get a 2 by 1 matrix showing the profit at each location.

Nick Johnson

Numerade Educator

Problem 86

Constructing a Cylindrical Tube A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Jason Taylor-Pestell

Jason Taylor-Pestell

Numerade Educator

Problem 86

Financial Planning A young couple has $\$ 25,000$ to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield $7 \%$, some money in corporate bonds that yield $9 \%$, and some money in junk bonds that yield $11 \%$. Prepare a table showing the various ways that this couple can achieve the following goals:
(a) $\$ 1500$ per year in income
(b) $\$ 2000$ per year in income
(c) $\$ 2500$ per year in income
(d) What advice would you give this couple regarding the income that they require and the choices available?
[Hint: Higher yields generally carry more risk.]

Christine Anacker

Christine Anacker

Numerade Educator

Problem 87

Cryptography One method of encryption is to use a matrix to encrypt the message and then use the corresponding inverse matrix to decode the message. The encrypted matrix, $E$, is obtained by multiplying the message matrix, $M$, by a key matrix, $K$. The original message can be retrieved by multiplying the encrypted matrix by the inverse of the key matrix. That is, $E=M \cdot K$ and $M=E \cdot K^{-1}$.
(a) Given the key matrix $K=\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{array}\right]$, find its inverse, $K^{-1}$. [Note: This key matrix is known as the $Q_2^3$ Fibonacci encryption matrix.]
(b) Use your result from part (a) to decode the encrypted
$$
\text { matrix } E=\left[\begin{array}{lll}
47 & 34 & 33 \\
44 & 36 & 27 \\
47 & 41 & 20
\end{array}\right] \text {. }
$$
(c) Each entry in your result for part (b) represents the position of a letter in the English alphabet $(A=1, B=2$, $C=3$, and so on). What is the original message?

David Mccaslin

Numerade Educator

Problem 87

Fencing A farmer has 300 feet of fence available to enclose a 4500 -square-foot region in the shape of adjoining squares, with sides of length $x$ and $y$. See the figure. Find $x$ and $y$.

Jason Taylor-Pestell

Jason Taylor-Pestell

Numerade Educator

Problem 87

Pharmacy A doctor's prescription calls for a daily intake of a supplement containing 40 milligrams $(\mathrm{mg}$ ) of vitamin $\mathrm{C}$ and $30 \mathrm{mg}$ of vitamin D. Your pharmacy stocks three supplements that can be used: one contains $20 \%$ vitamin $\mathrm{C}$ and $30 \%$ vitamin $D$; a second, $40 \%$ vitamin $C$ and $20 \%$ vitamin $D$; and a third, $30 \%$ vitamin $C$ and $50 \%$ vitamin D. Create a table showing the possible combinations that could be used to fill the prescription.

Christine Anacker

Christine Anacker

Numerade Educator

Problem 88

Economic Mobility The relative income of a child (low, medium, or high) generally depends on the relative income of the child's parents. The matrix $P$, given by
Parent's Income
$$
P=\left[\begin{array}{ccc}
\text { L } & \text { M } & \text { H } \\
0.4 & 0.2 & 0.1 \\
0.5 & 0.6 & 0.5 \\
0.1 & 0.2 & 0.4
\end{array}\right] \begin{array}{ll}
\mathrm{L} & \\
\mathrm{H} & \text { Child's income }
\end{array}
$$
is called a left stochastic transition matrix. For example, the entry $P_{21}=0.5$ means that $50 \%$ of the children of low relative income parents will transition to the medium level of income. The diagonal entry $P_{i i}$ represents the percent of children who remain in the same income level as their parents. Assuming that the transition matrix is valid from one generation to the next, compute and interpret $P^2$.
Source: Understanding Mobility in America, April 2006

Nick Johnson

Numerade Educator

Problem 88

Bending Wire A wire 60 feet long is cut into two pieces. Is it possible to bend one piece into the shape of a square and the other into the shape of a circle so that the total area enclosed by the two pieces is 100 square feet? If this is possible, find the length of the side of the square and the radius of the circle.

Prashant Bana

Numerade Educator

Problem 88

Pharmacy A doctor's prescription calls for the creation of pills that contain 12 units of vitamin $B_{12}$ and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains $20 \%$ vitamin $B_{12}$ and $30 \%$ vitamin $\mathrm{E}$; a second, $40 \%$ vitamin $\mathrm{B}_{12}$ and $20 \%$ vitamin $\mathrm{E}$; and a third, $30 \%$ vitamin $\mathrm{B}_{12}$ and $40 \%$ vitamin $\mathrm{E}$. Create a table showing the possible combinations of each powder that could be mixed in each pill.

Christine Anacker

Christine Anacker

Numerade Educator

Problem 89

Consider the 2 by 2 square matrix
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
If $D=a d-b c \neq 0$, show that $A$ is nonsingular and that
$$
A^{-1}=\frac{1}{D}\left[\begin{array}{rr}
d & -b \\
-c & a
\end{array}\right]
$$

Patrick Burns

Numerade Educator

Problem 89

Geometry Find formulas for the length $l$ and width $w$ of a rectangle in terms of its area $A$ and perimeter $P$.

Brittany Scott

Numerade Educator

Problem 89

Write a brief paragraph or two that outline your strategy for solving a system of linear equations using matrices.

Christine Anacker

Christine Anacker

Numerade Educator

Problem 90

Create a situation different from any found in the text that can be represented by a matrix.

Patrick Burns

Numerade Educator

Problem 90

Geometry Find formulas for the base $b$ and one of the equal sides $l$ of an isosceles triangle in terms of its altitude $h$ and perimeter $P$.

Pawan Yadav

Numerade Educator

Problem 90

When solving a system of linear equations using matrices, do you prefer to place the augmented matrix in row echelon form or in reduced row echelon form? Give reasons for your choice.

Christine Anacker

Christine Anacker

Numerade Educator

Problem 91

Explain why the number of columns in matrix $A$ must equal the number of rows in matrix $B$ when finding the product $A B$.

Patrick Burns

Numerade Educator

Problem 91

Descartes's Method of Equal Roots Descartes's method for finding tangents depends on the idea that, for many graphs, the tangent line at a given point is the unique line that intersects the graph at that point only. Apply his method to find an equation of the tangent line to the parabola $y=x^2$ at the point $(2,4)$. See the figure.
(GRAPH CANT COPY)
First, we know that the equation of the tangent line must be in the form $y=m x+b$. Using the fact that the point $(2,4)$ is on the line, we can solve for $b$ in terms of $m$ and get the equation $y=m x+(4-2 m)$. Now we want $(2,4)$ to be the unique solution to the system
$$
\left\{\begin{array}{l}
y=x^2 \\
y=m x+4-2 m
\end{array}\right.
$$
From this system, we get $x^2-m x+(2 m-4)=0$. By using the quadratic formula, we get
$$
x=\frac{m \pm \sqrt{m^2-4(2 m-4)}}{2}
$$
To obtain a unique solution for $x$, the two roots must be equal; in other words, the discriminant $m^2-4(2 m-4)$ must be 0 . Complete the work to get $m$, and write an equation of the tangent line.

Brittany Scott

Numerade Educator

Problem 91

Make up a system of three linear equations containing three variables that has:
(a) No solution
(b) Exactly one solution
(c) Infinitely many solutions
Give the three systems to a friend to solve and critique.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 92

If $a, b$, and $c \neq 0$ are real numbers with $a c=b c$, then $a=b$. Does this same property hold for matrices? In other words, if $A, B$, and $C$, are matrices and $A C=B C$, must $A=B$ ?

Patrick Burns

Numerade Educator

Problem 92

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$x^2+y^2=10$; at $(1,3)$

Ziya Ogron

Numerade Educator

Problem 93

What is the solution of the system of equations $A X=\varnothing$, if $A^{-1}$ exists? Discuss the solution of $A X=\varnothing$ if $A^{-1}$ does not exist.

Nick Johnson

Numerade Educator

Problem 93

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$y=x^2+2$; at $(1,3)$

Brittany Scott

Numerade Educator

Problem 94

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$x^2+y=5$; at $(-2,1)$

Brittany Scott

Numerade Educator

Problem 95

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$2 x^2+3 y^2=14$; at $(1,2)$

Pawan Yadav

Numerade Educator

Problem 96

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$3 x^2+y^2=7$; at $(-1,2)$

Pawan Yadav

Numerade Educator

Problem 97

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$x^2-y^2=3$; at $(2,1)$

Pawan Yadav

Numerade Educator

Problem 98

In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tangent to each graph at the given point.
$2 y^2-x^2=14$; at $(2,3)$

Pawan Yadav

Numerade Educator

Problem 99

If $r_1$ and $r_2$ are two solutions of a quadratic equation $a x^2+b x+c=0$, it can be shown that
$$
r_1+r_2=-\frac{b}{a} \text { and } r_1 r_2=\frac{c}{a}
$$
Solve this system of equations for $r_1$ and $r_2$.

Brittany Scott

Numerade Educator

Problem 100

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4 ? What about a polynomial of degree $n$ ? Can you explain your conclusions using an algebraic argument?

Pawan Yadav

Numerade Educator

Problem 100

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4 ? What about a polynomial of degree $n$ ? Can you explain your conclusions using an algebraic argument?

Pawan Yadav

Numerade Educator

Problem 101

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on top. The boxes are required to have a square
base and a 9-cubic-foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of sheet metal and folding along the edges.
(a) What are the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet?
(b) Could you make the box using a smaller piece of sheet metal? Make a list of the dimensions of the box for various pieces of sheet metal.

Derek Follett

Numerade Educator

Problem 387

In Problems 37-72, solve each system of equations using malrices (row operations). If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}x+2 y=5 \\ x+y=3\end{array}\right.$

Check back soon!

Problem 423

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{aligned} 2 x+y & =-4 \\ -2 y+4 z & =0 \\ 3 x-2 z & =-11\end{aligned}\right.$

Stephanie Carter

Stephanie Carter

Numerade Educator