Elementary and Intermediate Algebra 5th

Alan S. Tussy, R. David Gustafson

Chapter 3

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

Educators

Section 6

Point–Slope Form

Problem 1

Fill in the blanks.
$y-y_{1}=m\left(x-x_{1}\right)$ is called the _____ form of the equation of a line. In words, we read this as _____ $y$ minus $y$ _____ one equals $m$ _____ the quantity of $x$ _____ $x$ sub _____.

Julie S.

Julie S.

Numerade Educator

Problem 2

Fill in the blanks.
$y=m x+b$ is called the _____ form of the equation of a line.

Julie S.

Numerade Educator

Problem 3

Determine in what form each equation is written.
a. $y-4=2(x-5)$
b. $y=2 x+15$

Julie S.

Numerade Educator

Problem 4

What point does the graph of each equation pass through, and what is the line's slope?
a. $y-2=6(x-7)$
b. $y+3=-8(x+1)$

Julie S.

Numerade Educator

Problem 5

Refer to the following graph of a line
a. What highlighted point does the line pass through?
b. What is the slope of the line?
c. Write an equation of the line in point-slope form.
(GRAPH CANT COPY)

Julie S.

Numerade Educator

Problem 6

On a quiz, a student was asked to write the equation of a line with slope 4 that passes through $(-1,3) .$ Explain how the student can check her answer, $y=4 x+7$

Julie S.

Numerade Educator

Problem 7

Suppose you are asked to write an equation of the line in the scatter diagram below. What two points would you use to write the point-slope equation?
(DIAGRAM CANT COPY)

Julie S.

Numerade Educator

Problem 8

In each case, a linear relationship between two quantities is described. If the relationship were graphed, what would be the slope of the line?
a. The sales of new cars increased by 15 every 2 months.
b. There were 35 fewer robberies for each dozen police officers added to the force.
c. One acre of forest is being destroyed every 30 seconds.

Julie S.

Numerade Educator

Problem 9

Complete the solution.
Find an equation of the line with slope $-2$ that passes through the point $(-1,5) .$ Write the answer in slope-intercept form.
$$\begin{aligned} y-y_{1} &=m\left(x-x_{1}\right) \\ y-\quad &=-2[x-()] \\ y-5 &=-2[x-] \\ y-5 &=-2 x-\\ y &=-2 x+\end{aligned}$$

Julie S.

Numerade Educator

Problem 10

What is the point-slope form of the equation of a line?

Julie S.

Numerade Educator

Problem 11

Consider the steps below and then fill in the blanks:
$$
\begin{array}{c}
{y-3=2(x+1)} \\
{y-3=2 x+2} \\
{y=2 x+5}
\end{array}
$$
The original equation was in _____ form. After solving for $y,$ we obtain an equation in ______ form.

Julie S.

Numerade Educator

Problem 12

Fill in the blanks: The equation of a horizontal line has the form $\quad=b$ and the equation of a vertical line has the form $\quad=a$

Julie S.

Numerade Educator

Problem 13

Use the point-slope form to write an equation of the line with the given slope and point. Leave the equation in that form. See Example 1.
Slope $3,$ passes through $(2,1)$

Julie S.

Numerade Educator

Problem 14

Use the point-slope form to write an equation of the line with the given slope and point. Leave the equation in that form. See Example 1.
Slope $2,$ passes through $(4,3)$

Julie S.

Numerade Educator

Problem 15

Use the point-slope form to write an equation of the line with the given slope and point. Leave the equation in that form. See Example 1.
Slope $\frac{4}{5},$ passes through $(-5,-1)$

Julie S.

Numerade Educator

Problem 16

Use the point-slope form to write an equation of the line with the given slope and point. Leave the equation in that form. See Example 1.
Slope $\frac{7}{8},$ passes through $(-2,-9)$

Julie S.

Numerade Educator

Problem 17

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $2,$ passes through $(3,5)$

Julie S.

Numerade Educator

Problem 18

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $8,$ passes through $(2,6)$

Julie S.

Numerade Educator

Problem 19

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-5,$ passes through $(-9,8)$

Julie S.

Numerade Educator

Problem 20

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-4,$ passes through $(-2,10)$

Julie S.

Numerade Educator

Problem 21

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-3,$ passes through the origin

Julie S.

Numerade Educator

Problem 22

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-1,$ passes through the origin

Julie S.

Numerade Educator

Problem 23

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $\frac{1}{5},$ passes through $(10,1)$

Julie S.

Numerade Educator

Problem 24

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $\frac{1}{4},$ passes through $(8,1)$

Julie S.

Numerade Educator

Problem 25

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-\frac{4}{3}, \quad \begin{array}{|c|c|}\hline x & {y} \\ \hline 6 & {-4} \\ \hline\end{array}$

Julie S.

Numerade Educator

Problem 26

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-\frac{3}{2}, \quad \begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {1} \\ \hline\end{array}$

Julie S.

Numerade Educator

Problem 27

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-\frac{11}{6},$ passes through $(2,-6)$

Julie S.

Numerade Educator

Problem 28

Use the point-slope form to find an equation of the line with the given slope and point. Then write the equation in slope-intercept form. See Example $1 .$
Slope $-\frac{5}{4},$ passes through $(2,0)$

Julie S.

Numerade Educator

Problem 29

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(1,7)$ and $(-2,1)$

Julie S.

Numerade Educator

Problem 30

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(-2,2)$ and $(2,-8)$

Julie S.

Numerade Educator

Problem 31

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
$$
\begin{array}{|r|r|}
\hline x & {y} \\
\hline-4 & {3} \\
{2} & {0} \\
\hline
\end{array}
$$

Julie S.

Numerade Educator

Problem 32

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
$$
\begin{array}{|r|r|}
\hline x & {y} \\
\hline-1 & {-4} \\
{1} & {-2} \\
\hline
\end{array}
$$

Julie S.

Numerade Educator

Problem 33

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(5,5)$ and $(7,5)$

Julie S.

Numerade Educator

Problem 34

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(-2,1)$ and $(-2,15)$

Julie S.

Numerade Educator

Problem 35

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(5,1)$ and $(-5,0)$

Julie S.

Numerade Educator

Problem 36

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(-3,0)$ and $(3,1)$

Julie S.

Numerade Educator

Problem 37

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $(-8,2)$ and $(-8,17)$

Julie S.

Numerade Educator

Problem 38

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $\left(\frac{2}{3}, 2\right)$ and $(0,2)$

Julie S.

Numerade Educator

Problem 39

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $\left(\frac{2}{3}, \frac{1}{3}\right)$ and $(0,0)$

Julie S.

Numerade Educator

Problem 40

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2.
Passes through $\left(\frac{1}{2}, \frac{3}{4}\right)$ and $(0,0)$

Julie S.

Numerade Educator

Problem 41

Write an equation of each line. See Example 3.
Vertical, passes through $(4,5)$

Julie S.

Numerade Educator

Problem 42

Write an equation of each line. See Example 3.
Vertical, passes through $(-2,-5)$

Julie S.

Numerade Educator

Problem 43

Write an equation of each line. See Example 3.
Horizontal, passes through $(4,5)$

Julie S.

Numerade Educator

Problem 44

Write an equation of each line. See Example 3.
Horizontal, passes through $(-2,-5)$

Julie S.

Numerade Educator

Problem 45

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(1,-2), m=-1
$$

Julie S.

Numerade Educator

Problem 46

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(-4,1), m=-3
$$

Julie S.

Numerade Educator

Problem 47

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(5,-3), m=\frac{3}{4}
$$

Julie S.

Numerade Educator

Problem 48

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(2,-4), \quad m=\frac{2}{3}
$$

Julie S.

Numerade Educator

Problem 49

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(-2,-3), \text { slope } 2
$$

Julie S.

Numerade Educator

Problem 50

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(-3,-3), \text { slope } 4
$$

Julie S.

Numerade Educator

Problem 51

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(4,-3), \text { slope }-\frac{7}{8}
$$

Julie S.

Numerade Educator

Problem 52

Graph the line that passes through the given point and has the given slope. See Example $4 .$
$$
(4,2), \text { slope }-\frac{1}{5}
$$

Julie S.

Numerade Educator

Problem 53

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Passes through $(5,0)$ and $(-11,-4)$

Julie S.

Numerade Educator

Problem 54

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Passes through $(7,-3)$ and $(-5,1)$

Julie S.

Numerade Educator

Problem 55

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Horizontal, passes through $(-8,12)$

Julie S.

Numerade Educator

Problem 56

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Horizontal, passes through $(9,-32)$

Julie S.

Numerade Educator

Problem 57

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
$$
\text { Slope }-\frac{1}{4}, y \text { -intercept }\left(0, \frac{7}{8}\right)
$$

Julie S.

Numerade Educator

Problem 58

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $-\frac{9}{5}, y$ -intercept $\left(0, \frac{11}{3}\right)$

Julie S.

Numerade Educator

Problem 59

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $-\frac{2}{3},$ passes through $(3,0)$

Julie S.

Numerade Educator

Problem 60

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $-\frac{2}{5},$ passes through $(15,0)$

Julie S.

Numerade Educator

Problem 61

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $8,$ passes through $(2,20)$

Julie S.

Numerade Educator

Problem 62

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $6,$ passes through $(1,-2)$

Julie S.

Numerade Educator

Problem 63

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Vertical, passes through $(-3,7)$

Julie S.

Numerade Educator

Problem 64

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Vertical, passes through $(12,-23)$

Julie S.

Numerade Educator

Problem 65

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope 7 and $y$ -intercept $(0,-11)$

Julie S.

Numerade Educator

Problem 66

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope 3 and $y$ -intercept $(0,4)$

Julie S.

Numerade Educator

Problem 67

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Passes through $(-2,-1)$ and $(-1,-5)$

Julie S.

Numerade Educator

Problem 68

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Passes through $(-3,6)$ and $(-1,-4)$

Julie S.

Numerade Educator

Problem 69

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
$x$ -intercept $(7,0)$ and $y$ -intercept $(0,-2)$

Julie S.

Numerade Educator

Problem 70

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
$x$ -intercept $(-3,0)$ and $y$ -intercept $(0,7)$

Julie S.

Numerade Educator

Problem 71

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $\frac{1}{10},$ passes through the origin

Julie S.

Numerade Educator

Problem 72

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $\frac{9}{8},$ passes through the origin

Julie S.

Numerade Educator

Problem 73

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Undefined slope, passes through $\left(-\frac{1}{8}, 12\right)$

Julie S.

Numerade Educator

Problem 74

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Undefined slope, passes through $\left(\frac{2}{5},-\frac{5}{6}\right)$

Julie S.

Numerade Educator

Problem 75

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $1.7, y$ -intercept $(0,-2.8)$

Julie S.

Numerade Educator

Problem 76

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible.
Slope $9.5, y$ -intercept $(0,-14.3)$

Julie S.

Numerade Educator

Problem 77

There is a linear relationship between a woman's height and the length of her radius bone. It can be stated this way:
Height increases by 3.9 inches for each 1-inch increase in the length of the radius. Suppose a 64 -inch-tall woman has a 9 -inch-long radius bone. Use this information to find a linear equation that relates height $h$ to the length $r$ of the radius. Write the equation in slope-intercept form.

Julie S.

Numerade Educator

Problem 78

An automated production line uses distilled water at a rate of 300 gallons every 2 hours to make shampoo. After the line had run for 7 hours, planners noted that $2,500$ gallons of distilled water remained in the storage tank. Find a linear equation relating the time $t$ in hours since the production line began and the number $g$ of gallons of distilled water in the storage tank. Write the equation in slope-intercept form.

Julie S.

Numerade Educator

Problem 79

Find the equations of the lines that describe the positions of the pole for parts $1,3,$ and 4 of the jump. Write the equations in slope-intercept form, if possible.
(GRAPH CANT COPY)

Julie S.

Numerade Educator

Problem 80

The graph below shows the route of a proposed freeway.
a. Give the coordinates of the points where the proposed Freeway 133 will join Interstate 25 and Highway $40 .$
b. Find an equation of the line that describes the route of the proposed freeway. Write the equation in slope-intercept form.
(GRAPH CANT COPY)

Julie S.

Numerade Educator

Problem 81

Three months after cleanup began at a dump site, 800 cubic yards of toxic waste had yet to be removed. Two months later, that number had been lowered to 720 cubic yards.
a. Find an equation that describes the linear relationship between the length of time $m$ (in months) the cleanup crew has been working and the number of cubic yards $y$ of toxic waste remaining. Write the equation in slope-intercept form.
b. Use your answer to part (a) to predict the number of cubic yards of waste that will still be on the site one year after the cleanup project began.

Julie S.

Numerade Educator

Problem 82

To lower its corporate income tax, accountants of a company depreciated a word processing system over several years using a linear model, as shown in the worksheet.
a. Find a linear equation relating the years since the system was purchased, $x,$ and its value, $y,$ in dollars. Write the equation in slope-intercept form.
b. Find the purchase price of the system.

Julie S.

Numerade Educator

Problem 83

There is a linear relationship between the length of the protective pad that wraps around a trampoline and the radius of the trampoline. Use the data in the table to find an equation that gives the length $l$ of pad needed for any trampoline with radius $r .$ Write the equation in slope-intercept form. Use units of feet for both $l$ and $r$
$$
\begin{array}{|c|c|}
\hline \text { Radius } & {\text { Pad length }} \\
\hline 3 \mathrm{ft} & {19 \mathrm{ft}} \\
{7 \mathrm{ft}} & {44 \mathrm{ft}} \\
\hline
\end{array}
$$

Julie S.

Numerade Educator

Problem 84

In the report "Expenditures on Children by Families," the U.S. Department of Agriculture projected the yearly child-rearing expenditures on children from birth through age $17 .$ For a child born in 2010 to a two-parent middle-income family, the report estimated annual expenditures of $\$ 10,808$ when the child is 6 years old, and $\$ 14,570$ when the child is 15 years old.
a. Write two ordered pairs of the form (child's age, annual expenditure).
b. Assume the relationship between the child's age $a$ and the annual expenditures $E$ is linear. Use your answers to part (a) to write an equation in slope-intercept form that models this relationship.
c. What are the projected child-rearing expenses when the child is 17 years old?

Julie S.

Numerade Educator

Problem 85

The scatter diagram shows the amount of milk that an average American drank in one year for the years $1980-2008 .$ A straight line can be used to model the data.
a. Use the two highlighted points on the line to find its equation. Write the equation in slope-intercept form.
b. Use your answer to part (a) to predict the amount of milk that an average American will drink in 2020 .
(DIAGRAM CANT COPY)

Julie S.

Numerade Educator

Problem 86

The horsepower produced by an automobile engine was recorded for various engine speeds in the range of $2,400$ to $4,800$ revolutions per minute (rpm). The data were recorded on the following scatter diagram. Find an equation of the line that models the relationship between engine speed $s$ and horsepower $h .$ Write the equation in slope-intercept form.
(DIAGRAM CANT COPY)

Julie S.

Numerade Educator

Problem 87

Why is $y-y_{1}=m\left(x-x_{1}\right)$ called the point-slope form of the equation of a line?

Julie S.

Numerade Educator

Problem 88

If we know two points that a line passes through, we can write its equation. Explain how this is done.

Julie S.

Numerade Educator

Problem 89

Explain the steps involved in writing $y-6=4(x-1)$ in slope-intercept form.

Julie S.

Numerade Educator

Problem 90

Think of the points on the graph of the horizontal line $y=4$ What do the points have in common? How do they differ?

Julie S.

Numerade Educator

Problem 91

The length of a rectangular picture is 5 inches greater than twice the width. If the perimeter is 112 inches, find the dimensions of the frame.

Julie S.

Numerade Educator

Problem 92

Two planes are $6,000$ miles apart, and their speeds differ by $200 \mathrm{mph}$. They travel toward each other and meet in 5 hours. Find the speed of the slower plane.

Julie S.

Numerade Educator

Problem 93

Find an equation of the line that passes through $(2,5)$ and is parallel to the line $y=4 x-7 .$ Write the equation in slope-intercept form.

Julie S.

Numerade Educator

Problem 94

Find an equation of the line that passes through $(-6,3)$ and is perpendicular to the line $y=-3 x-12 .$ Write the equation in slope-intercept form.

Julie S.

Numerade Educator