Systems of equations and inequalities | Practice Problems, Examples & Solutions

Linear Systems of Equations and Inequalities with Two Variables

691 Practice Problems

02:16

Algebra and Trigonometry

Solve each system by substitution.
$$\begin{array}{r}
x+3 y=5 \\
2 x+3 y=4
\end{array}$$

Systems of Equations and Inequalities

Systems of Linear Equations: Two Variables

01:37

Precalculus

So the input (3,5) yields an output of $\sqrt{2} .$ We define the do-
main for this function just as we did in Chapter 3: The domain is the set of all inputs that yield real-number outputs. For instance, the ordered pair (1,4) is not in the domain of the function we have been discussing, because (as you should check for yourself$) f(1,4)=\sqrt{-1},$ which is not a real number. We can determine the domain of the function in equation ( 1 ) by requiring that the quantity under the radical sign be non negative. Thus we require that $2 x-y+1 \geq 0$ and, consequently, $y \leq 2 x+1$ (Check this.) The following figure shows the graph of this inequality; the domain of our function is the set of ordered pairs making up the graph. In Exercises follow a similar procedure and sketch the domain of the given function.
(Graph cant copy)
$$f(x, y)=\sqrt{x^{2}+y^{2}-1}$$

Systems of Equations

Systems of Inequalities

02:54

Precalculus

Graph the following system of inequalities and specify the vertices.
$$\left\{\begin{array}{l}
x \geq 0 \\
y \geq e^{x} \\
y \leq e^{-x}+1
\end{array}\right.$$
A formula such as
$$f(x, y)=\sqrt{2 x-y+1}$$
defines a function of two variables. The inputs for such a function are ordered pairs $(x, y)$ of real numbers. For example, using the ordered pair (3,5) as an input, we have
$$f(3,5)=\sqrt{2(3)-5+1}=\sqrt{2}$$

Systems of Equations

Systems of Inequalities

Linear Systems of Equations with Three Variables

234 Practice Problems

04:24

Precalculus

Some applications of systems lead to systems similar to those that follow. Solve using elimination.
$$\left\{\begin{aligned} -2 A-B-3 C &=21 \\ B-C &=1 \\ A+B &=-4 \end{aligned}\right.$$

Systems of Equations and Inequalities

Linear Systems in Three Variables with Applications

06:52

Precalculus

Solve using the elimination method. If a system is inconsistent or dependent, so state. For systems with linear dependence, write the answer in terms of a parameter. For coincident dependence, state the solution in set notation.
$$\left\{\begin{array}{c} 4 x-5 y-6 z=5 \\ 2 x-3 y+3 z=0 \\ x+2 y-3 z=5 \end{array}\right.$$

Systems of Equations and Inequalities

Linear Systems in Three Variables with Applications

08:13

Precalculus

Solve using elimination. If the system is linearly dependent, state the general solution in terms of a parameter. Different forms of the solution are possible.
$$\left\{\begin{array}{l} 3 x-4 y+5 z=5 \\ -x+2 y-3 z=-3 \\ 3 x-2 y+z=1 \end{array}\right.$$

Systems of Equations and Inequalities

Linear Systems in Three Variables with Applications

Nonlinear Systems of Equations and Inequalities with Two Variables

310 Practice Problems

01:36

Introductory and Intermediate Algebra for College Students

Solve the systems in Exercises $79-80$.
$$\left\{\begin{array}{l}
\log _{y} x=3 \\
\log _{y}(4 x)=5
\end{array}\right.$$

Conic Sections and Systems of Nonlinear Equations

Systems of Nonlinear Equations in Two Variables

00:42

Introductory and Intermediate Algebra for College Students

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
A system of two equations in two variables whose graphs are two circles must have at least two real ordered-pair solutions.

Conic Sections and Systems of Nonlinear Equations

Systems of Nonlinear Equations in Two Variables

02:16

Introductory and Intermediate Algebra for College Students

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I think that the nonlinear system consisting of $x^{2}+y^{2}=36$ and $y=(x-2)^{2}-3$ is easier to solve graphically than by using the substitution method or the addition method.

Conic Sections and Systems of Nonlinear Equations

Systems of Nonlinear Equations in Two Variables

Solving Systems of Equations by Graphing

65 Practice Problems

02:00

Algebra and Trigonometry Real Mathematics, Real People

Write an equation of the line passing through the two points. Use the slope-intercept form, if possible. If not possible, explain why.
$$\left(\frac{3}{5}, 0\right),(4,6)$$

Linear Systems and Matrices

Solving Systems of Equations

01:36

Algebra and Trigonometry Real Mathematics, Real People

Write an equation of the line passing through the two points. Use the slope-intercept form, if possible. If not possible, explain why.
$$(6,3),(10,3)$$

Linear Systems and Matrices

Solving Systems of Equations

03:04

Algebra and Trigonometry Real Mathematics, Real People

Determine whether the statement is true or false. Justify your answer.
If a system consists of a parabola and a circle, then it can have at most two solutions.

Linear Systems and Matrices

Solving Systems of Equations

Solving Systems of Equations Algebraically

68 Practice Problems

02:06

Algebra and Trigonometry Real Mathematics, Real People

You are offered two jobs selling college textbooks. One company offers an annual salary of $\$ 33,000$ plus a year-end bonus of $1 \%$ of your total sales. The other company offers an annual salary of $\$ 30,000$ plus a year-end bonus of $2.5 \%$ of your total sales. How much would you have to sell in a year to make the second offer the better offer?

Linear Systems and Matrices

Solving Systems of Equations

02:01

Algebra and Trigonometry Real Mathematics, Real People

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales $x$ necessary to break even $(R=C)$ and the corresponding revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
Cost
$C=5.5 \sqrt{x}+10,000$

Revenue
$R=3.29 x$

Linear Systems and Matrices

Solving Systems of Equations

03:07

Algebra and Trigonometry Real Mathematics, Real People

Solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{aligned}
2 \ln x+y &=4 \\
e^{x}-y &=0
\end{aligned}\right.$$

Linear Systems and Matrices

Solving Systems of Equations

Solving Systems of Equations in Three Variables

12 Practice Problems

01:29

Prealgebra and Introductory Algebra

Rewrite equation so that the coefficients are integers. Then solve the system of equations by the substitution method.
$0.8 x-0.1 y=0.3$
$0.5 x-0.2 y=-0.5$

Systems of Linear Equations

Solving Systems of Linear Equations by the Substitution Method

01:17

Prealgebra and Introductory Algebra

Assume that $A, B$, and $C$ are nonzero real numbers. State whether the system of equations is independent, inconsistent, or dependent.
$$\begin{aligned}x+y &=B \\y &=-x+C, C \neq B\end{aligned}$$

Systems of Linear Equations

Solving Systems of Linear Equations by the Substitution Method

01:00

Prealgebra and Introductory Algebra

Assume that $A, B,$ and $C$ are nonzero real numbers, where $A \neq B \neq C .$ State whether the system of equations is independent, inconsistent, or dependent.
$$\begin{array}{r}
A x+B y=C \\
2 A x+2 B y=2 C
\end{array}$$

Systems of Linear Equations

Solving Systems of Linear Equations by the Addition Method

Solving Systems of Equations Using Determinants

28 Practice Problems

06:11

Algebra and Trigonometry Real Mathematics, Real People

(A) find the determinant of $A,$ (b) find $A^{-1},$ (c) find $\operatorname{det}\left(A^{-1}\right),$ and
(d) compare your results from parts (a) and (c). Make a conjecture based on your results.
$$A=\left[\begin{array}{rrr}
-1 & 3 & 2 \\
1 & 3 & -1 \\
1 & 1 & -2
\end{array}\right]$$

Linear Systems and Matrices

The Determinant of a Square Matrix

00:32

Algebra and Trigonometry Real Mathematics, Real People

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{ll}
x & \ln x \\
1 & 1 / x
\end{array}\right|$$

Linear Systems and Matrices

The Determinant of a Square Matrix

01:40

Algebra and Trigonometry Real Mathematics, Real People

Solve for $x$
$$\left|\begin{array}{rrr}
1 & 2 & x \\
-1 & 3 & 2 \\
3 & -2 & 1
\end{array}\right|=0$$

Linear Systems and Matrices

The Determinant of a Square Matrix

Problem Solving Using Systems of Two Equations

13 Practice Problems

04:00

Prealgebra and Introductory Algebra

Write three different systems of equations: a. one that has $(-3,5)$ as its only solution, b. one for which there is no solution, and c. one that is a dependent system of equations.

Systems of Linear Equations

Solving Systems of Linear Equations by Graphing

01:14

Prealgebra and Introductory Algebra

A, B, C,$ and $D$ are nonzero real numbers. State whether the system of equations is independent, inconsistent, or dependent.
$$\begin{aligned}
&x=C\\
&y=D
\end{aligned}$$

Systems of Linear Equations

Solving Systems of Linear Equations by Graphing

00:37

Prealgebra and Introductory Algebra

Determine whether the statement is always true, sometimes true, or never true.
The system of two linear equations graphed at the right has no solution.
(THE GRAPH CANNOT COPY)

Systems of Linear Equations

Solving Systems of Linear Equations by Graphing

Problem Solving Using Systems of Three Equations

1 Practice Problems

02:23

Intermediate Algebra

Write a system of three equations in three variables that models the situation. Do not solve the system.
A bakery makes three kinds of pies: chocolate cream, which sells for $\$ 5$; apple, which sells for $\$ 6$; and cherry, which sells for $\$ 7 .$ The cost to make the pies is $\$ 2, \$ 3,$ and $\$ 4$ respectively. Let $x=$ the number of chocolate cream pies made daily, $y=$ the number of apple pies made daily, and $z=$ the number of cherry pies made daily.
-Each day, the bakery makes 50 pies.
-Each day, the revenue from the sale of the pies is $\$ 295$.
-Each day, the cost to make the pies is $\$ 145 .$

Systems of Equations

Problem Solving Using Systems of Three Equations