Intermediate Algebra

Alan S. Tussy, R. David Gustafson

Chapter 3

Systems of Equations - all with Video Answers

Educators

Section 1

Solving Systems of Equations by Graphing

Problem 1

Fill in the blanks.
$\left\{\begin{array}{l}x-2 y=4 \\ 2 x-y=3\end{array}\right.$ is called a _____ of linear equations.

Joseph Lentino

Joseph Lentino

Numerade Educator

Problem 2

Fill in the blanks.
When a system of equations has at least one solution, it is called a ____ system. If a system has no solutions, it is called an ____ system.

Joseph Lentino

Numerade Educator

Problem 3

Fill in the blanks.
If two equations have different graphs, they are called ____ equations. Two equations with the same graph are called ____ equations.

Joseph Lentino

Numerade Educator

Problem 4

Fill in the blanks.
When solving a system of two linear equations by the graphing method, we look for the point of ____ of the two lines.

Joseph Lentino

Numerade Educator

Problem 5

Refer to the illustration. Decide whether a true or a false statement would be obtained
a. when the coordinates of Point $A$ are substituted into the equation for line $l_{1}$
b. when the coordinates of Point $B$ are substituted into the equation for line $l_{1}$
c. when the coordinates of Point $C$ are substituted into the equation for line $l_{1}$
d. when the coordinates of Point $C$ are substituted into the equation for line $l_{2}$
(GRAPH NOT COPY)

Joseph Lentino

Numerade Educator

Problem 6

Refer to the illustration.
a. How many ordered pairs satisfy the equation $3 x+y=3 ?$ Name three.
b. How many ordered pairs satisfy the equation $\frac{2}{3} x-y=-3 ?$ Name three.
c. How many ordered pairs satisfy both equations? Name it or them.
(GRAPH NOT COPY)

Joseph Lentino

Numerade Educator

Problem 7

a. How many solutions does the system of equations graphed on the right have? Are the equations dependent or independent?
b. How many solutions does the system of equations graphed on the right have? Give three of the solutions. Is the system consistent or inconsistent?
(GRAPH NOT COPY)

Joseph Lentino

Numerade Educator

Problem 8

Estimate the solution of the system of linear equations shown in the following display.
(TABLE NOT COPY)

Joseph Lentino

Numerade Educator

Problem 9

Use the graphs in the illustration below to solve each equation.
(GRAPH NOT COPY)
$$
-3 x+2=x-2
$$

Joseph Lentino

Numerade Educator

Problem 10

Use the graphs in the illustration below to solve each equation.
(GRAPH NOT COPY)
$$
-x-4=x-2
$$

Joseph Lentino

Numerade Educator

Problem 11

Fill in the blanks.
The symbol $\{$ is called a left ____. It is used when writing a system of equations.

Joseph Lentino

Numerade Educator

Problem 12

Fill in the blanks.
We read the set-builder notation $\{(x, y) | 3 x-5 y=1\}$ as "the ____ of all ordered pairs $(x, y)$ ____ that $3 x-5 y=1$.

Joseph Lentino

Numerade Educator

Problem 13

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(-4,3) ;\left\{\begin{array}{l}
4 x-y=-19 \\
3 x+2 y=-6
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 14

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(-1,2) ;\left\{\begin{array}{l}
3 x-y=-5 \\
x-y=-4
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 15

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(2,-3) ;\left\{\begin{array}{l}
y+2=\frac{1}{2} x \\
3 x+2 y=0
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 16

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(1,2) ;\left\{\begin{array}{l}
2 x-y=0 \\
y=\frac{1}{2} x+\frac{3}{2}
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 17

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
\left(\frac{1}{2}, \frac{1}{3}\right) ;\left\{\begin{array}{l}
2 x+3 y=2 \\
4 x-9 y=1
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 18

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
\left(-\frac{3}{4}, \frac{2}{3}\right) ;\left\{\begin{array}{l}
4 x+3 y=-1 \\
4 x-3 y=-5
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 19

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(-0.2,0.5) ;\left\{\begin{array}{l}
2 x+5 y=2.1 \\
5 x+y=-0.5
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 20

Determine whether the ordered pair is a solution of the system of equations. See Example 1.
$$
(2.1,-3.2) ;\left\{\begin{array}{c}
x+y=-1.1 \\
2 x-3 y=13.8
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 21

Solve each system by graphing. See Example 2.
$$
\left\{\begin{array}{l}
x+y=6 \\
x-y=2
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 22

Solve each system by graphing. See Example 2.
$$
\left\{\begin{array}{l}
x-y=4 \\
2 x+y=5
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 23

Solve each system by graphing. See Example 2.
$$
\left\{\begin{array}{l}
y=-2 x+1 \\
x-2 y=-7
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 24

Solve each system by graphing. See Example 2.
$$
\left\{\begin{array}{l}
3 x-y=-3 \\
y=-2 x-7
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 25

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4
$$
\left\{\begin{array}{l}
3 x-3 y=4 \\
x-y=4
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 26

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4.
$$
\left\{\begin{array}{l}
5 x+2 y=6 \\
-10 x-4 y=-12
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 27

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4.
$$
\left\{\begin{array}{l}
x=3-2 y \\
2 x+4 y=6
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 28

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4.
$$
\left\{\begin{array}{l}
3 x=5-2 y \\
3 x+2 y=7
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 29

Solve each system by graphing. See Example 5 .
$$
\left\{\begin{array}{l}
\frac{1}{6} x=\frac{1}{3} y+\frac{1}{2} \\
y=x
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 30

Solve each system by graphing. See Example 5 .
$$
\left\{\begin{array}{l}
x=y+3 \\
\frac{1}{4} x-\frac{1}{6} y=\frac{1}{3}
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 31

Solve each system by graphing. See Example 5 .
$$
\left\{\begin{array}{l}
\frac{1}{3} x-\frac{7}{6} y=\frac{1}{2} \\
\frac{1}{5} y=\frac{1}{3} x+\frac{7}{15}
\end{array}\right.
$$

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 32

Solve each system by graphing. See Example 5 .
$$
\left\{\begin{array}{l}
\frac{3}{5} x+\frac{1}{4} y=-\frac{11}{10} \\
\frac{1}{8} x=\frac{13}{24}+\frac{1}{3} y
\end{array}\right.
$$

Stepfan Ng

Numerade Educator

Problem 33

Solve each equation graphically. See Example $6 .$
$$
2 x+1=5
$$

Joseph Lentino

Numerade Educator

Problem 34

Solve each equation graphically. See Example $6 .$
$$
3 x+5=-4
$$

Joseph Lentino

Numerade Educator

Problem 35

Solve each equation graphically. See Example $6 .$
$$
-2 x+8=3 x-7
$$

Joseph Lentino

Numerade Educator

Problem 36

Solve each equation graphically. See Example $6 .$
$$
2 x-3=3 x-3
$$

Joseph Lentino

Numerade Educator

Problem 37

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing.
$$
\left\{\begin{array}{l}
y=3.2 x-1.5 \\
y=-2.7 x-3.7
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 38

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing.
$$
\left\{\begin{array}{l}
y=-0.45 x+5 \\
y=5.55 x-13.7
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 39

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing.
$$
\left\{\begin{array}{l}
1.7 x+2.3 y=3.2 \\
y=0.25 x+8.95
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 40

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing.
$$
\left\{\begin{array}{l}
2.75 x=12.9 y-3.79 \\
7.1 x-y=35.76
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 41

Use a graphing calculator to solve each equation.
$$
4 x-4=3 x
$$

Joseph Lentino

Numerade Educator

Problem 42

Use a graphing calculator to solve each equation.
$$
4(x-3)-x=x-6
$$

Joseph Lentino

Numerade Educator

Problem 43

Use a graphing calculator to solve each equation.
$$
11 x+6(3-x)=3
$$

Joseph Lentino

Numerade Educator

Problem 44

Use a graphing calculator to solve each equation.
$$
2 x+4=12-2 x
$$

Joseph Lentino

Numerade Educator

Problem 45

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
y=-\frac{5}{2} x+\frac{1}{2} \\
2 x-\frac{3}{2} y=5
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 46

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
y=-\frac{5}{6} x+2 \\
\frac{5}{2} x+3 y=6
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 47

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x+y=0 \\
y=2 x-6
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 48

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
3 x+y=3 \\
3 x+2 y=0
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 49

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
y=3 \\
x=2
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 50

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
2 x+3 y=-15 \\
2 x+y=-9
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 51

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=\frac{11-2 y}{3} \\
y=\frac{11-6 x}{4}
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 52

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=\frac{1-3 y}{4} \\
y=\frac{12+3 x}{2}
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 53

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
4 x-3 y=5 \\
y=-2 x
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 54

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
2 x+2 y=-1 \\
3 x+4 y=0
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 55

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=13-4 y \\
3 x=4+2 y
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 56

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
3 x=7-2 y \\
2 x=2+4 y
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 57

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=2 \\
y=-\frac{1}{2} x+2
\end{array}\right.
$$

FK

Fatih Kilinc

Numerade Educator

Problem 58

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
y=-2 \\
y=\frac{2}{3} x-\frac{4}{3}
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 59

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=\frac{5}{2} y-2 \\
x-\frac{5}{3} y+\frac{1}{3}=0
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 60

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
2 x=5 y-11 \\
3 x=2 y
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 61

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x+3 y=6 \\
y=-\frac{1}{3} x+2
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 62

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
2 x-y=-4 \\
2 y=4 x-6
\end{array}\right.
$$

Cory Kuzinski

Numerade Educator

Problem 63

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
x=-\frac{3}{2} y \\
2 x=3 y-4
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 64

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.)
$$
\left\{\begin{array}{l}
4 x=3 y-1 \\
3 y=4-8 x
\end{array}\right.
$$

Joseph Lentino

Numerade Educator

Problem 65

One of the line graphs below gives the percent share of the U.S. footwear market for shoes produced in the United States. The other line gives the percent share of the U.S. footwear market for imports.
a. Estimate the coordinates of the point of intersection of the graphs.
b. What important percent-of-the-market information does your answer to part a give?
(GRAPH NOT COPY)

Joseph Lentino

Numerade Educator

Problem 66

Business. Estimate the break-even point (where cost $=$ revenue) on the graph in the illustration. Explain why it is called the break-even point.
(GRAPH NOT COPY)

Joseph Lentino

Numerade Educator

Problem 67

Hearing Tests. See the illustration in the next column. At what frequency and decibel level were the hearing test results the same for the left and right ear? Write your answer as an ordered pair.
(GRAPH NOT COPY)

Jerelyn Nevil

Numerade Educator

Problem 68

The graph below shows the growing importance of the Internet in the daily lives of Americans. Determine when the time spent on the following activities was the same. Approximately how many hours per year were spent on each?
a. Internet and reading magazines
b. Internet and reading newspapers
c. Internet and reading books
(GRAPH NOT COPY)

Jerelyn Nevil

Numerade Educator

Problem 69

The demand function, graphed below, describes the relationship between the price $x$ of a certain camera and the demand for the camera.
a. The supply function, $S(x)=\frac{25}{4} x-525,$ describes the relationship between the price $x$ of the camera and the number of cameras the manufacturer is willing to supply. Graph this function in the illustration.
b. For what price will the supply of cameras equal the demand?
c. As the price of the camera is increased, what happens to supply and what happens to demand?
(GRAPH NOT COPY)

Jerelyn Nevil

Numerade Educator

Problem 70

The function $C(x)=200 x+400$ gives the cost for a college to offer $x$ sections of an introductory class in CPR (cardiopulmonary resuscitation). The function $R(x)=280 x$ gives the amount of revenue the college brings in when offering $x$ sections of CPR.
a. Find the break-even point (where cost = revenue) by graphing each function on the same coordinate system.
b. How many sections does the college need to offer to make a profit on the CPR training course?

Cory Kuzinski

Numerade Educator

Problem 71

The paths of two ships are tracked on the same coordinate system. One ship is following a path described by the equation $2 x+3 y=6,$ and the other is following a path described by the equation $y=\frac{2}{3} x-3$
a. Graph both equations on the same coordinate system. Is there
a possibility of a collision?
b. Is a collision a certainty?

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 72

Two airplanes, flying at the same altitude, are tracked using the same coordinate system on a radar screen. One plane is following a path described by the equation $y=0.4 x-2,$ and the other is following a path described by the equation $2 x=5 y+7 .$ Graph both equations on the same coordinate system. Is there a possibility of a collision?

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 73

Suppose the solution of a system of two linear equations is $\left(\frac{14}{5},-\frac{8}{3}\right) .$ Knowing this, explain any drawbacks you might encounter when solving the system by the graphing method.

Joseph Lentino

Numerade Educator

Problem 74

Can a system of two linear equations have exactly two solutions? Why or why not?

Joseph Lentino

Numerade Educator

Problem 75

Suppose the graphs of the two linear equations of a system are the same line. What is wrong with this statement? The system has infinitely many solutions. Any ordered pair is a solution of the system.

Ian Maurer

Numerade Educator

Problem 76

a. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=3 x+2$ intersect?
b. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=2 x+2$ do not intersect?
c. Without graphing, how can you tell that the graphs of $y=2 x+3$ and $2 y=4 x+6$ are the same line?

Joseph Lentino

Numerade Educator

Problem 77

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
f(-1)
$$

Joseph Lentino

Numerade Educator

Problem 78

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
f(10)
$$

Joseph Lentino

Numerade Educator

Problem 79

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
g(2)
$$

Joseph Lentino

Numerade Educator

Problem 80

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
g(-20)
$$

Joseph Lentino

Numerade Educator

Problem 81

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
f(t)
$$

Joseph Lentino

Numerade Educator

Problem 82

Let $f(x)=-x^{3}+2 x-2$ and $g(x)=\frac{2-x}{9+x}$ and find each value.
$$
g(s+1)
$$

Joseph Lentino

Numerade Educator

Problem 83

Find the domain of the function $f(x)=\frac{1}{3 x+6}$

Joseph Lentino

Numerade Educator

Problem 84

Find the domain of the function $f(x)=|x|$

Joseph Lentino

Numerade Educator

Problem 85

Write a dependent system of equations with a solution of $(-5,2)$

Joseph Lentino

Numerade Educator

Problem 86

Write a system of three independent linear equations in two variables that has the solution $(-5,2)$

Joseph Lentino

Numerade Educator