Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer

Chapter 4

Systems of Linear Equations and Inequalities - all with Video Answers

Educators

Section 1

Solving Systems of Linear Equations by Graphing

Problem 1

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(2,-3)\\
&\left\{\begin{array}{l}
2 x+3 y=-5 \\
7 x-3 y=23
\end{array}\right.
\end{aligned}$$

John Irizar

John Irizar

Numerade Educator

Problem 2

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(-2,-5)\\
&\left\{\begin{array}{l}
6 x-2 y=-2 \\
3 x+y=-11
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 3

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&\left(\frac{2}{3}, \frac{1}{9}\right)\\
&\left\{\begin{array}{r}
x+3 y=1 \\
4 x+3 y=3
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 4

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&\left(\frac{7}{25},-\frac{1}{25}\right)\\
&\left\{\begin{array}{l}
4 x+3 y=1 \\
3 x-4 y=1
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 5

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(-5,9)\\
&\left\{\begin{array}{r}
5 x+3 y=2 \\
x+4 y=14
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 6

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(10,7)\\
&\left\{\begin{array}{l}
6 x-5 y=25 \\
4 x+15 y=13
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 7

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(1400,450)\\
&\left\{\begin{aligned}
x-2 y &=500 \\
0.03 x+0.02 y &=51
\end{aligned}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 8

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(200,700)\\
&\left\{\begin{array}{c}
-4 x+y=-100 \\
0.05 x-0.06 y=-32
\end{array}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 9

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(8,5)\\
&\left\{\begin{aligned}
5 x-4 y &=20 \\
3 y &=2 x+1
\end{aligned}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 10

In Exercises $1-10$, determine whether the given ordered pair is a solution of the system.
$$\begin{aligned}
&(5,-2)\\
&\left\{\begin{aligned}
4 x-3 y &=26 \\
x &=15-5 y
\end{aligned}\right.
\end{aligned}$$

John Irizar

Numerade Educator

Problem 11

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=6 \\
x-y=2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 12

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=2 \\
x-y=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 13

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=1 \\
y-x=3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 14

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=4 \\
y-x=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 15

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
2 x-3 y=6 \\
4 x+3 y=12
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 16

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+2 y=2 \\
x-y=2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 17

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
4 x+y=4 \\
3 x-y=3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 18

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
5 x-y=10 \\
2 x+y=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 19

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=x+5 \\
y=-x+3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 20

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=x+1 \\
y=3 x-1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 21

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=2 x \\
y=-x+6
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 22

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=2 x+1 \\
y=-2 x-3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 23

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=-2 x+3 \\
y=-x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 24

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=3 x-4 \\
y=-2 x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 25

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=2 x-1 \\
y=2 x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 26

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=3 x-1 \\
y=3 x+2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 27

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=4 \\
x=-2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 28

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=6 \\
y=-3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 29

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x-2 y=4 \\
2 x-4 y=8
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 30

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
2 x+3 y=6 \\
4 x+6 y=12
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 31

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=2 x-1 \\
x-2 y=-4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 32

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=-2 x-4 \\
4 x-2 y=8
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 33

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x+y=5 \\
2 x+2 y=12
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 34

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x-y=2 \\
3 x-3 y=-6
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 35

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x-y=0 \\
y=x
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 36

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
2 x-y=0 \\
y=2 x
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 37

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x=2 \\
y=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 38

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x=3 \\
y=5
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 39

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x=2 \\
x=-1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 40

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x=3 \\
x=-2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 41

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=0 \\
y=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 42

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
y=0 \\
y=5
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 43

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
y=\frac{1}{2} x-3 \\
y=\frac{1}{2} x-5
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 44

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
y=\frac{3}{4} x-2 \\
y=\frac{3}{4} x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 45

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
y=-\frac{1}{2} x+4 \\
3 x-y=-4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 46

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
y=-\frac{1}{4} x+3 \\
4 x-y=-3
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 47

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
3 x-y=6 \\
x=\frac{y}{3}+2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 48

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
2 x-y=4 \\
x=\frac{y}{2}+2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 49

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
3 x+y=0 \\
y=-3 x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 50

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
2 x+y=0 \\
y=-2 x+1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 51

A rental company charges $\$ 40.00$ a day plus $\$ 0.35$ per mile to rent a moving truck. The total cost, $y,$ for a day's rental if $x$ miles are driven is described by $y=0.35 x+40 . A$ second company charges $\$ 36.00$ a day plus $\$ 0.45$ per mile, so the daily cost, $y,$ if $x$ miles are driven is described by $y=0.45 x+36 .$ The graphs of the two equations are shown in the same rectangular coordinate system.
(GRAPH NOT COPY)
a. What is the $x$ -coordinate of the intersection point of the graphs? Describe what this $x$ -coordinate means in practical terms.
b. What is a reasonable cstimate for the $y$ -coordinate of the intersection point?
c. Substitute the $x$ -coordinate of the intersection point into each of the equations and find the corresponding value for $y .$ Describe what this value represents in practical terms. How close is this value to your estimate from part (b)?

John Irizar

Numerade Educator

Problem 52

A band plans to record a demo. Studio A rents for $\$ 100$ plus $\$ 50$ per hour. Studio $B$ rents for $\$ 50$ plus $\$ 75$ per hour. The total cost, $y,$ in dollars, of renting the studios for $x$ hours can be modeled by the linear system
a. Use graphing to solve the system. Extend the $x$ -axis from 0 to 4 and let each tick mark represent 1 unit (one hour in a recording studio). Extend the $y$ -axis from 0 to 400 and let each tick mark represent 100 units (a rental $\operatorname{cost} \text { of } \$ 100)$
b. Interpret the coordinates of the solution in practical terms.

John Irizar

Numerade Educator

Problem 53

You plan to start taking an aerobics class. Nonmembers pay $\$ 4$ per class. Members pay a $\$ 10$ monthly fee plus an additional $\$ 2$ per class. The monthly cost, $y,$ of taking $x$ classes can be modeled by the linear system
a. Use graphing to solve the system.
b. Interpret the coordinates of the solution in practical terms.

John Irizar

Numerade Educator

Problem 54

What is a system of linear equations? Provide an example with your description.

John Irizar

Numerade Educator

Problem 55

What is a solution of a system of linear equations?

John Irizar

Numerade Educator

Problem 56

Explain how to determine if an ordered pair is a solution of a system of linear equations.

John Irizar

Numerade Educator

Problem 57

Explain how to solve a system of linear equations by graphing.

John Irizar

Numerade Educator

Problem 58

What is an inconsistent system? What happens if you attempt to solve such a system by graphing?

John Irizar

Numerade Educator

Problem 59

Explain how a linear system can have infinitely many solutions.

John Irizar

Numerade Educator

Problem 60

What are dependent equations? Provide an example with your description.

John Irizar

Numerade Educator

Problem 61

The following system models the winning times, $y,$ in seconds, in the Olympic 500 -meter speed skating event
$x$ years after 1970 :
\left\{\begin{array}{l}
y=-0.19 x+43.7 \\
y=-0.16 x+39.9
\end{array}\right.
Use the slope of each model to explain why the system has a solution. What does this solution represent?

John Irizar

Numerade Educator

Problem 62

Make Sense? In Exercises $62-65$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
Each equation in a system of linear equations has infinitely many ordered-pair solutions.

John Irizar

Numerade Educator

Problem 63

In Exercises $62-65$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
Every linear system has infinitely many ordered-pair solutions.

John Irizar

Numerade Educator

Problem 64

In Exercises $62-65$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
In dependent systems, the two equations represent the same line.

John Irizar

Numerade Educator

Problem 65

In Exercises $62-65$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
When I use graphing to solve an inconsistent system, the lines should look parallel, and I can always use slope to confirm that they really are.

John Irizar

Numerade Educator

Problem 66

In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If a linear system has graphs with equal slopes, the system must be inconsistent.

John Irizar

Numerade Educator

Problem 67

In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If a linear system has graphs with equal $y$ -intercepts, the system must have infinitely many solutions.

John Irizar

Numerade Educator

Problem 68

In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If a linear system has two distinct points that are solutions, then the graphs of the system's equations have equal slopes and equal $y$ -intercepts.

John Irizar

Numerade Educator

Problem 69

In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
It is possible for a linear system with one solution to have graphs with equal slopes.

John Irizar

Numerade Educator

Problem 70

Write a system of linear equations whose solution is $(5,1)$ How many different systems are possible? Explain.

John Irizar

Numerade Educator

Problem 71

Write a system of equations with one solution, a system of equations with no solution, and a system of equations with infinitely many solutions. Explain how you were able to think of these systems.

John Irizar

Numerade Educator

Problem 72

Verify your solutions to any five exercises from Exercises 11 through 36 by using a graphing utility to graph the two equations in the system in the same viewing rectangle. After entering the two equations, one as $y_{1}$ and the other as $y_{2},$ and graphing them, use the $[\text { NTERSECTION }]$ feature to find the system's solution. (It may first be necessary to solve the equations for $y$ before entering them.)

John Irizar

Numerade Educator

Problem 73

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
y=2 x+2 \\
y=-2 x+6
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 74

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
y=-x+5 \\
y=x-7
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 75

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
x+2 y=4 \\
x-y=4
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 76

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
2 x-3 y=10 \\
4 x+3 y=20
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 77

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{c}
3 x-y=5 \\
-5 x+2 y=-10
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 78

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
2 x-3 y=7 \\
3 x+5 y=1
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 79

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
y=\frac{1}{3} x+\frac{2}{3} \\
y=\frac{5}{7} x-2
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 80

Read Exercise 72. Then use a graphing utility to solve the systems in Exercises $73-80$
$$\left\{\begin{array}{l}
y=-\frac{1}{2} x+2 \\
y=\frac{3}{4} x+7
\end{array}\right.$$

John Irizar

Numerade Educator

Problem 81

In Exercises $81-83$, perform the indicated operation.
$-3+(-9)$ (Section $1.7,$ Table 1.7 )

John Irizar

Numerade Educator

Problem 82

In Exercises $81-83$, perform the indicated operation.
$-3-(-9)$ (Section $1.7,$ Table 1.7 )

John Irizar

Numerade Educator

Problem 83

In Exercises $81-83$, perform the indicated operation.
$-3(-9)$ (Section $1.7,$ Table 1.7 )

John Irizar

Numerade Educator

Problem 84

Exercises $84-86$ will help you prepare for the material covered in the next section. In each exercise, solve the given equation.
$$4 x-3(-x-1)=24$$

John Irizar

Numerade Educator

Problem 85

Exercises $84-86$ will help you prepare for the material covered in the next section. In each exercise, solve the given equation.
$$5(2 y-3)-4 y=9$$

John Irizar

Numerade Educator

Problem 86

Exercises $84-86$ will help you prepare for the material covered in the next section. In each exercise, solve the given equation.
$$(5 x-1)+1=5 x+5$$

John Irizar

Numerade Educator