Introductory Algebra for College Students

Robert Blitzer

Chapter 3

Linear Equations and Inequalities in Two Variables - all with Video Answers

Educators

ES

Section 1

Graphing Linear Equations in Two Variables

Problem 1

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(3,5)$$

Kristen Karbon

Kristen Karbon

Numerade Educator

Problem 2

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(5,3)$$

Rebecca Wang

Numerade Educator

Problem 3

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(-5,1)$$

Kristen Karbon

Numerade Educator

Problem 4

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(1,-5)$$

Rebecca Wang

Numerade Educator

Problem 5

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(-3,-1)$$

Kristen Karbon

Numerade Educator

Problem 6

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(-1,-3)$$

Rebecca Wang

Numerade Educator

Problem 7

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(6,-3.5)$$

Kristen Karbon

Numerade Educator

Problem 8

plot the given point in a rectangular coordinate system. Indicate in which quadrant each point lies.
$$(-3.5,6)$$

Rebecca Wang

Numerade Educator

Problem 9

plot the given point in a rectangular coordinate system.
$$(-3,-3)$$

Kristen Karbon

Numerade Educator

Problem 10

plot the given point in a rectangular coordinate system.
$$(-5,-5)$$

Rebecca Wang

Numerade Educator

Problem 11

plot the given point in a rectangular coordinate system.
$$(-2,0)$$

Kristen Karbon

Numerade Educator

Problem 12

plot the given point in a rectangular coordinate system.
$$(-5,0)$$

Rebecca Wang

Numerade Educator

Problem 13

plot the given point in a rectangular coordinate system.
$$(0,2)$$

Kristen Karbon

Numerade Educator

Problem 14

plot the given point in a rectangular coordinate system.
$$(0,5)$$

Rebecca Wang

Numerade Educator

Problem 15

plot the given point in a rectangular coordinate system.
$$(0,-3)$$

Kristen Karbon

Numerade Educator

Problem 16

plot the given point in a rectangular coordinate system.
$$(0,-5)$$

Rebecca Wang

Numerade Educator

Problem 17

plot the given point in a rectangular coordinate system.
$$\left(\frac{5}{2}, \frac{7}{2}\right)$$

Kristen Karbon

Numerade Educator

Problem 18

plot the given point in a rectangular coordinate system.
$$\left(\frac{7}{2}, \frac{5}{2}\right)$$

Rebecca Wang

Numerade Educator

Problem 19

plot the given point in a rectangular coordinate system.
$$\left(-5, \frac{3}{2}\right)$$

Kristen Karbon

Numerade Educator

Problem 20

plot the given point in a rectangular coordinate system.
$$\left(-\frac{9}{2},-4\right)$$

Rebecca Wang

Numerade Educator

Problem 21

plot the given point in a rectangular coordinate system.
$$(0,0)$$

Kristen Karbon

Numerade Educator

Problem 22

plot the given point in a rectangular coordinate system.
$$\left(-\frac{5}{2}, 0\right)$$

Rebecca Wang

Numerade Educator

Problem 23

plot the given point in a rectangular coordinate system.
$$\left(0,-\frac{5}{2}\right)$$

Kristen Karbon

Numerade Educator

Problem 24

plot the given point in a rectangular coordinate system.
$$\left(0, \frac{7}{2}\right)$$

Rebecca Wang

Numerade Educator

Problem 25

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
A

Julie Silva

Numerade Educator

Problem 26

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
B

Rebecca Wang

Numerade Educator

Problem 27

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
C

Matthias Wuest

Numerade Educator

Problem 28

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
D

Matthias Wuest

Numerade Educator

Problem 29

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
E

Matthias Wuest

Numerade Educator

Problem 30

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
F

Rebecca Wang

Numerade Educator

Problem 31

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
G

Matthias Wuest

Numerade Educator

Problem 32

give the ordered pairs that correspond to the points labeled in the figure.(graph can't copy)
H

Matthias Wuest

Numerade Educator

Problem 33

In which quadrants are the $y$ -coordinates positive?

Kristen Karbon

Numerade Educator

Problem 34

In which quadrants are the $x$ -coordinates negative?

Rebecca Wang

Numerade Educator

Problem 35

In which quadrants do the $x$ -coordinates and the $y$ -coordinates have the same sign?

Kristen Karbon

Numerade Educator

Problem 36

In which quadrants do the $x$ -coordinates and the $y$ -coordinates have opposite signs?

Rebecca Wang

Numerade Educator

Problem 37

determine whether each ordered pair is a solution of the given equation.
$$y=3 x \quad(2,3),(3,2),(-4,-12)$$

Kristen Karbon

Numerade Educator

Problem 38

determine whether each ordered pair is a solution of the given equation.
$$y=4 x \quad(3,12),(12,3),(-5,-20)$$

Rebecca Wang

Numerade Educator

Problem 39

determine whether each ordered pair is a solution of the given equation.
$$y=-4 x \quad(-5,-20),(0,0),(9,-36)$$

Kristen Karbon

Numerade Educator

Problem 40

determine whether each ordered pair is a solution of the given equation.
$$y=-3 x \quad(-5,15),(0,0),(7,-21)$$

Rebecca Wang

Numerade Educator

Problem 41

determine whether each ordered pair is a solution of the given equation.
$$y=2 x+6 \quad(0,6),(-3,0),(2,-2)$$

Kristen Karbon

Numerade Educator

Problem 42

determine whether each ordered pair is a solution of the given equation.
$$y=8-4 x \quad(8,0),(16,-2),(3,-4)$$

Rebecca Wang

Numerade Educator

Problem 43

determine whether each ordered pair is a solution of the given equation.
$$3 x+5 y=15 \quad(-5,6),(0,5),(10,-3)$$

Kristen Karbon

Numerade Educator

Problem 44

determine whether each ordered pair is a solution of the given equation.
$$2 x-5 y=0 \quad(-2,0),(-10,6),(5,0)$$

Rebecca Wang

Numerade Educator

Problem 45

determine whether each ordered pair is a solution of the given equation.
$$x+3 y=0 \quad(0,0),\left(1, \frac{1}{3}\right),\left(2,-\frac{2}{3}\right)$$

Kristen Karbon

Numerade Educator

Problem 46

determine whether each ordered pair is a solution of the given equation.
$$x+5 y=0 \quad(0,0),\left(1, \frac{1}{5}\right),\left(2,-\frac{2}{5}\right)$$

Rebecca Wang

Numerade Educator

Problem 47

determine whether each ordered pair is a solution of the given equation.
$$x-4=0 \quad(4,7),(3,4),(0,-4)$$

Rebecca Wang

Numerade Educator

Problem 48

determine whether each ordered pair is a solution of the given equation.
$$y+2=0 \quad(0,2),(2,0),(0,-2)$$

Rebecca Wang

Numerade Educator

Problem 49

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=12 x$$

Kristen Karbon

Numerade Educator

Problem 50

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=14 x$$

Rebecca Wang

Numerade Educator

Problem 51

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=-10 x$$

Kristen Karbon

Numerade Educator

Problem 52

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=-20 x$$

Rebecca Wang

Numerade Educator

Problem 53

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=8 x-5$$

Kristen Karbon

Numerade Educator

Problem 54

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=6 x-4$$

Rebecca Wang

Numerade Educator

Problem 55

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=-3 x+7$$

Kristen Karbon

Numerade Educator

Problem 56

find five solutions of each equation. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Organize your work in a table of values.
$$y=-5 x+9$$

Rebecca Wang

Numerade Educator

Problem 57

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x$$

Kristen Karbon

Numerade Educator

Problem 58

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x+1$$

Rebecca Wang

Numerade Educator

Problem 59

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x-1$$

Kristen Karbon

Numerade Educator

Problem 60

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x-2$$

Rebecca Wang

Numerade Educator

Problem 61

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=2 x+1$$

Kristen Karbon

Numerade Educator

Problem 62

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=2 x-1$$

Rebecca Wang

Numerade Educator

Problem 63

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-x+2$$

Kristen Karbon

Numerade Educator

Problem 64

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-x+3$$

Rebecca Wang

Numerade Educator

Problem 65

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-3 x-1$$

Kristen Karbon

Numerade Educator

Problem 66

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-3 x-2$$

Rebecca Wang

Numerade Educator

Problem 67

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=\frac{1}{2} x$$

Rebecca Wang

Numerade Educator

Problem 68

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{1}{2} x$$

Kristen Karbon

Numerade Educator

Problem 69

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{1}{4} x$$

Rebecca Wang

Numerade Educator

Problem 70

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=\frac{1}{4} x$$

Kristen Karbon

Numerade Educator

Problem 71

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=\frac{1}{3} x+1$$

Kristen Karbon

Numerade Educator

Problem 72

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=\frac{1}{3} x-1$$

Rebecca Wang

Numerade Educator

Problem 73

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{3}{2} x+1$$

Kristen Karbon

Numerade Educator

Problem 74

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{3}{2} x+2$$

Rebecca Wang

Numerade Educator

Problem 75

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{5}{2} x-1$$

Kristen Karbon

Numerade Educator

Problem 76

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=-\frac{5}{2} x+1$$

Rebecca Wang

Numerade Educator

Problem 77

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x+\frac{1}{2}$$

Kristen Karbon

Numerade Educator

Problem 78

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=x-\frac{1}{2}$$

Rebecca Wang

Numerade Educator

Problem 79

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation.
$$y=4, \text { or } y=0 x+4$$

Kristen Karbon

Numerade Educator

Problem 80

write each sentence as a linear equation in two variables. Then graph the equation.
$$y=3, \text { or } y=0 x+3$$

Matthias Wuest

Numerade Educator

Problem 81

write each sentence as a linear equation in two variables. Then graph the equation.
The $y$ -variable is 3 more than the $x$ -variable.

Kristen Karbon

Numerade Educator

Problem 82

write each sentence as a linear equation in two variables. Then graph the equation.

The $y$ -variable exceeds the $x$ -variable by 4

Rebecca Wang

Numerade Educator

Problem 83

write each sentence as a linear equation in two variables. Then graph the equation.
The $y$ -variable exceeds twice the $x$ -variable by 5

Kristen Karbon

Numerade Educator

Problem 84

write each sentence as a linear equation in two variables. Then graph the equation.
The $y$ -variable is 2 less than 3 times the $x$ -variable.

Rebecca Wang

Numerade Educator

Problem 85

At the beginning of a semester, a student purchased cight pens and six pads for a total cost of $\$ 14.50 .$
a. If $x$ represents the cost of one pen and $y$ represents the cost of one pad, write an equation in two variables that reflects the given conditions.
b. If pads cost $\$ 0.75$ each, find the cost of one pen.

Matthias Wuest

Numerade Educator

Problem 86

A nursery offers a package of three small orange trees and four small grapefruit trees for $\$ 22$.
a. If $x$ represents the cost of one orange tree and $y$ represents the cost of one grapefruit tree, write an equation in two variables that reflects the given conditions.
b. If a grapefruit tree costs $\$ 2.50,$ find the cost of an orange tree.

Matthias Wuest

Numerade Educator

Problem 87

A football is thrown by a quarterback to a receiver. The points in the figure show the height of the football, in feet, above the ground in terms of its distance, in yards, from the quarterback. Use this information
Find the coordinates of point A. Then interpret the coordinates in terms of the information given.

Matthias Wuest

Numerade Educator

Problem 88

be sure to refer to the figure at the bottom of the previous page.)
Find the coordinates of point $B .$ Then interpret the coordinates in terms of the information given.

Matthias Wuest

Numerade Educator

Problem 89

be sure to refer to the figure at the bottom of the previous page.)
Estimate the coordinates of point $C$.

Matthias Wuest

Numerade Educator

Problem 90

be sure to refer to the figure at the bottom of the previous page.)
Estimate the coordinates of point $D$

Matthias Wuest

Numerade Educator

Problem 91

be sure to refer to the figure at the bottom of the previous page.)
What is the football's maximum height? What is its distance from the quarterback when it reaches its maximum height?

Matthias Wuest

Numerade Educator

Problem 92

be sure to refer to the figure at the bottom of the previous page.)
What is the football's height when it is caught by the receiver? What is the receiver's distance from the
quarterback when he catches the football?

Matthias Wuest

Numerade Educator

Problem 93

The graph shows that in $2000,31 \%$ of U.S. adults viewed
a college education as essential for success. For the period from 2000 through 2010 , the percentage viewing a college cducation as essential for success increased on average by approximately 2.4 each year. These conditions can be described by the mathematical model
$$
S=2.4 n+31
$$
where $S$ is the percentage of U.S. adults who vicwed college as essential for success $n$ years after 2000 .
a. Let $n=0,5,10,15,$ and $20 .$ Make a table of values showing five solutions of the equation.
b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled $n,$ represent 5 units. Extend the horizontal axis to include $n=25 .$ Let each tick mark on the vertical axis, labeled $S$, represent 10 units and extend the axis to include $S=100$

Matthias Wuest

Numerade Educator

Problem 94

The graph shows that in $2000,45 \%$ of U.S. adults believed that most qualified students get to attend college. For the period from 2000 through 2010 , the percentage who believed that a college education is available to most qualified students decreased by approximately 1.7 each year. These
- conditions can be described by the mathematical model
$$
Q=-1.7 n+45
$$
where $Q$ is the percentage believing that a college
- education is available to most qualificd students $n$ years after 2000
a. Let $n=0,5,10,15,$ and $20 .$ Make a table of values showing five solutions of the equation.
b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled $n$, represent 5 units. Extend the horizontal axis to include $n=25 .$ Let each tick mark on the vertical axis, labeled $Q$, represent 5 units and extend the axis to include $Q=50$
- c. Use your graph from part (b) to estimate the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 .
- d. Use the formula to project the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 .

Matthias Wuest

Numerade Educator

Problem 95

What is the rectangular coordinate system?

Elizabeth Xu

Numerade Educator

Problem 96

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

Elizabeth Xu

Numerade Educator

Problem 97

Explain why $(5,-2)$ and $(-2,5)$ do not represent the same point.

Kristen Karbon

Numerade Educator

Problem 98

Explain how to find the coordinates of a point in the rectangular coordinate system.

Rebecca Wang

Numerade Educator

Problem 99

How do you determine whether an ordered pair is a solution of an equation in two variables, $x$ and $y ?$

Kristen Karbon

Numerade Educator

Problem 100

Explain how to find ordered pairs that are solutions of an equation in two variables, $x$ and $y$

Rebecca Wang

Numerade Educator

Problem 101

What is the graph of an equation?

Kristen Karbon

Numerade Educator

Problem 102

Explain how to graph an cquation in two variables in the rectangular coordinate system.

Matthias Wuest

Numerade Educator

Problem 103

determine whether each statement “makes sense” or “does not make sense” and explain your reasoning.
When I know that an equation's graph is a straight line, I don't need to plot more than two points, although I sometimes plot three just to check that the points line up.

Kristen Karbon

Numerade Educator

Problem 104

determine whether each statement “makes sense” or “does not make sense” and explain your reasoning.
The graph that I'm looking at is U-shaped, so its equation cannot be of the form $y=m x+b$

Rebecca Wang

Numerade Educator

Problem 105

determine whether each statement “makes sense” or “does not make sense” and explain your reasoning.
I'm working with a linear equation in two variables and found that $(-2,2),(0,0),$ and $(2,2)$ are solutions.

Matthias Wuest

Numerade Educator

Problem 106

I'm working with a linear equation in two variables and found that $(-2,2),(0,0),$ and $(2,2)$ are solutions.
When a real-world situation is modeled with a linear cquation in two variables, I can use its graph to predict specific information about the situation.

Matthias Wuest

Numerade Educator

Problem 107

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of $y=3 x+1$ is the graph of $y=2 x$ shifted up 1 unit

Kristen Karbon

Numerade Educator

Problem 108

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of any equation in the form $y=m x+b$ passes through the point $(0, b)$

Rebecca Wang

Numerade Educator

Problem 109

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The ordered pair $(3,4)$ satisfies the equation
$$
2 y-3 x=-6
$$

Kristen Karbon

Numerade Educator

Problem 110

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $(2,5)$ satisfies an equation, then $(5,2)$ also satisfies the equation.

Rebecca Wang

Numerade Educator

Problem 111

a. Graph each of the following points:
$$
\left(1, \frac{1}{2}\right),(2,1),\left(3, \frac{3}{2}\right),(4,2)
$$
Parts (b)-(d) can be answered by changing the sign of one or both coordinates of the points in part (a).
b. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the $y$ -axis of your graph in part (a)?
c. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the $x$ -axis of your graph in part (a)?
d. What must be done to the coordinates so that the resulting graph is a straight-line extension of your graph in part (a)?

Kristen Karbon

Numerade Educator

Problem 112

Use a graphing utility to graph in a standard viewing rectangle, $[-10,10,1]$ by $[-10,10,1]$. Then use the $[\text { TRACE }]$ feature to trace along the line and find the coordinates of two points.
$$y=2 x-1$$

Jose Avalos

Numerade Educator

Problem 113

Use a graphing utility to graph in a standard viewing rectangle, $[-10,10,1]$ by $[-10,10,1]$. Then use the $[\text { TRACE }]$ feature to trace along the line and find the coordinates of two points.
$$y=-3 x+2$$

Kristen Karbon

Numerade Educator

Problem 114

Use a graphing utility to graph in a standard viewing rectangle, $[-10,10,1]$ by $[-10,10,1]$. Then use the $[\text { TRACE }]$ feature to trace along the line and find the coordinates of two points.
$$y=\frac{1}{2} x$$

Jose Avalos

Numerade Educator

Problem 115

Use a graphing utility to graph in a standard viewing rectangle, $[-10,10,1]$ by $[-10,10,1]$. Then use the $[\text { TRACE }]$ feature to trace along the line and find the coordinates of two points.
$$y=\frac{3}{4} x-2$$

Kristen Karbon

Numerade Educator

Problem 116

Use a graphing utility to verify any five of your hand-drawn graphs in Exercises $57-80 .$ Use an appropriate viewing rectangle and the $[\text { ZOOM SQUARE }]$ feature to make the graph look like the one you drew by hand.

Jose Avalos

Numerade Educator

Problem 117

$$\text { Solve: } 3 x+5=4(2 x-3)+7$$

ES

Emily Stein

Numerade Educator

Problem 118

$$\text { Simplify: } 3(1-2 \cdot 5)-(-28)$$

Matthias Wuest

Numerade Educator

Problem 119

Solve for $h: \quad V=\frac{1}{3} A h .$ (Section 2.4, Example 4)

Kristen Karbon

Numerade Educator

Problem 120

will help you prepare for the material covered in the next section. Remember that a solution of an
equation in two variables is an ordered pair.
Let $y=0$ and find a solution of $3 x-4 y=24$

Kristen Karbon

Numerade Educator

Problem 121

will help you prepare for the material covered in the next section. Remember that a solution of an
equation in two variables is an ordered pair.
Let $x=0$ and find a solution of $3 x-4 y=24$

Rebecca Wang

Numerade Educator

Problem 122

will help you prepare for the material covered in the next section. Remember that a solution of an
equation in two variables is an ordered pair.
Let $x=0$ and find a solution of $x+2 y=0$

Rebecca Wang

Numerade Educator