Beginning and Intermediate Algebra

Sherri Messersmith

Chapter 4

Linear Equations in Two Variables - all with Video Answers

Educators

Section 1

Introduction to Linear Equations in Two Variables

Problem 1

The graph shows the number of gallons of diet soda consumed per person for the years $2002-2006$. (U.S. Dept of Agriculture)
How many gallons of diet soda were consumed per person in $2005 ?$

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Problem 2

The graph shows the number of gallons of diet soda consumed per person for the years $2002-2006$. (U.S. Dept of Agriculture)
During which year was the consumption level about 15.0 gallons per person?

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Problem 3

The graph shows the number of gallons of diet soda consumed per person for the years $2002-2006$. (U.S. Dept of Agriculture)
During which two years was consumption the same, and how much diet soda was consumed each of these years?

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Problem 4

The graph shows the number of gallons of diet soda consumed per person for the years $2002-2006$. (U.S. Dept of Agriculture)
During which year did people drink the least amount of diet soda?

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Problem 5

During which year did people drink the least amount of diet soda?
What was the general consumption trend from 2002 to $2005 ?$

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Problem 6

The graph shows the number of gallons of diet soda consumed per person for the years $2002-2006$. (U.S. Dept of Agriculture)
Compare the consumption level in 2002 with that in 2006 .

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Problem 7

The bar graph shows the public high school graduation rate in certain states in 2006 .
Which state had the highest graduation rate, and what percentage of its public high school students graduated?

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Problem 8

The bar graph shows the public high school graduation rate in certain states in 2006 .
Which states graduated between $70 \%$ and $80 \%$ of its students?

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Problem 9

The bar graph shows the public high school graduation rate in certain states in 2006 .
How does the graduation rate of Florida compare with that of New Jersey?

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Problem 10

The bar graph shows the public high school graduation rate in certain states in 2006 .
Which state had a graduation rate of about $62 \% ?$

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Problem 11

The bar graph shows the public high school graduation rate in certain states in 2006 .
Explain the difference between a linear equation in one variable and a linear equation in two variables. Give an example of each.

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Problem 12

The bar graph shows the public high school graduation rate in certain states in 2006 .
True or False: $3 x+6 y^{2}=-1$ is a linear equation in two variables.

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Problem 13

Determine whether each ordered pair is a solution of the given equation.
$$
2 x+5 y=1 ;(-2,1)
$$

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Problem 14

Determine whether each ordered pair is a solution of the given equation.
$$
2 x+7 y=-4 ;(2,-5)
$$

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Problem 15

Determine whether each ordered pair is a solution of the given equation.
$$
-3 x-2 y=-15 ;(7,-3)
$$

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Problem 16

Determine whether each ordered pair is a solution of the given equation.
$$
y=5 x-6 ;(3,9)
$$

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Problem 17

Determine whether each ordered pair is a solution of the given equation.
$$
y=-\frac{3}{2} x-7 ;(8,5)
$$

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Problem 18

Determine whether each ordered pair is a solution of the given equation.
$$
5 y=\frac{2}{3} x+1 ;(6,1)
$$

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Problem 19

Determine whether each ordered pair is a solution of the given equation.
$$
y=-7 ;(9,-7)
$$

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Problem 20

Determine whether each ordered pair is a solution of the given equation.
$$
x=8 ;(-10,8)
$$

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Problem 21

Complete the ordered pair for each equation.
$$
y=3 x-7 ;(4, \quad)
$$

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Problem 22

Complete the ordered pair for each equation.
$$
y=-2 x+3 ;(6, \quad)
$$

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Problem 23

Complete the ordered pair for each equation.
$$
2 x-15 y=13 ;\left(\quad,-\frac{4}{3}\right)
$$

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Problem 24

Complete the ordered pair for each equation.
$$
-x+10 y=8 ;\left(\frac{2}{5}\right)
$$

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Problem 25

Complete the ordered pair for each equation.
$$
x=5 ;(\quad,-200)
$$

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Problem 26

Complete the ordered pair for each equation.
$$
y=-10 ;(12, \quad)
$$

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Problem 27

Complete the table of values for each equation.
$$
\begin{aligned}
&y=2 x-4\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline 1 & \\
\hline-1 & \\
\hline-2 & \\
\hline
\end{array}
\end{aligned}
$$

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Problem 28

Complete the table of values for each equation.
$$
\begin{aligned}
&y=-5 x+1\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline 1 & \\
\hline 2 & \\
\hline-1 &
\end{array}
\end{aligned}
$$

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Problem 29

Complete the table of values for each equation.
$$
\begin{aligned}
&y=4 x\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline \frac{1}{2} & \\
\hline & 12 \\
\hline & -20 \\
\hline
\end{array}
\end{aligned}
$$

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Problem 30

Complete the table of values for each equation.
$$
\begin{aligned}
&y=9 x-8\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline-\frac{1}{3} & \\
\hline & -17 \\
\hline & 1 \\
\hline
\end{array}
\end{aligned}
$$

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Problem 31

Complete the table of values for each equation.
$$
\begin{aligned}
&5 x+4 y=-8\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline & 0 \\
\hline 1 & \\
\hline-\frac{12}{5} & \\
\hline
\end{array}
\end{aligned}
$$

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Problem 32

Complete the table of values for each equation.
$$
\begin{aligned}
&2x-y=12\\
&\begin{array}{c|c}
x & y \\
\hline &0 \\
\hline 0 & \\
\hline &-2 \\
\hline\frac{5}{2}\\
\end{array}
\end{aligned}
$$

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Problem 33

Complete the table of values for each equation.
$$
\begin{aligned}
&y=-2\\
&\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & \\
\hline-3 & \\
\hline 8 & \\
\hline 17 & \\
\hline
\end{array}
\end{aligned}
$$

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Problem 34

Complete the table of values for each equation.
$$
\begin{aligned}
&x=20\\
&\begin{array}{c|c}
x & y \\
\hline & 0 \\
\hline & 3 \\
\hline & -4 \\
\hline & -9 \\
\hline
\end{array}
\end{aligned}
$$

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Problem 35

Explain, in words, how to complete the table of values for $x=-13$.
$$
\begin{array}{c|c}
\boldsymbol{x} & \boldsymbol{y} \\
\hline & 0 \\
\hline & 2 \\
\hline & -1
\end{array}
$$

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Problem 36

Explain, in words, how to complete the ordered pair $(\quad,-3)$ for $y=-x-2$

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Problem 37

Name each point with an ordered pair, and identify the quadrant in which each point lies.

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Problem 38

Name each point with an ordered pair, and identify the quadrant in which each point lies.

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Problem 39

Graph each ordered pair and explain how you plotted the points.
$$
(2,4)
$$

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Problem 40

Graph each ordered pair and explain how you plotted the points.
$$
(4,1)
$$

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Problem 41

Graph each ordered pair and explain how you plotted the points.
$$
(-3,-5)
$$

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Problem 42

Graph each ordered pair and explain how you plotted the points.
$$
(-2,1)
$$

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Problem 43

Graph each ordered pair.
$$
(-6,1)
$$

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Problem 44

Graph each ordered pair.
$$
(-2,-3)
$$

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Problem 45

Graph each ordered pair.
$$
(0,-1)
$$

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Problem 46

Graph each ordered pair.
$$
(4,-5)
$$

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Problem 47

Graph each ordered pair.
$$
(0,4)
$$

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Problem 48

Graph each ordered pair.
$$
(-5,0)
$$

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Problem 49

Graph each ordered pair.
$$
(-2,0)
$$

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Problem 50

Graph each ordered pair.
$$
(0,-1)
$$

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Problem 51

Graph each ordered pair.
$$
\left(-2, \frac{3}{2}\right)
$$

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Problem 52

Graph each ordered pair.
$$
\left(\frac{4}{3}, 3\right)
$$

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Problem 53

Graph each ordered pair.
$$
\left(3,-\frac{1}{4}\right)
$$

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Problem 54

Graph each ordered pair.
$$
\left(-2,-\frac{9}{4}\right)
$$

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Problem 55

Graph each ordered pair.
$$
\left(0,-\frac{11}{5}\right)
$$

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Problem 56

Graph each ordered pair.
$$
\left(-\frac{9}{2},-\frac{2}{3}\right)
$$

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Problem 57

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=-4 x+3\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline 2 & 0 \\
\hline & 7 \\
\hline
\end{array}
\end{aligned}
$$

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Problem 58

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=-3 x+4\\
&\begin{array}{c|c}
\boldsymbol{x} & \boldsymbol{y} \\
\hline 0 & \\
\hline & 0 \\
\hline & -2 \\
\hline-1 & \\
\hline
\end{array}
\end{aligned}
$$

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Problem 59

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=x\\
&\begin{array}{c|c}
\boldsymbol{x} & \boldsymbol{y} \\
\hline 0 & \\
\hline-1 & \\
\hline & 3 \\
\hline & -5
\end{array}
\end{aligned}
$$

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Problem 60

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=-2 x\\
&\begin{array}{c|c}
x & y \\
\hline & 0 \\
\hline & 4 \\
\hline 2 & \\
\hline-3 &
\end{array}
\end{aligned}
$$

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Problem 61

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&3 x+4 y=12\\
&\begin{array}{c|c}
x & y \\
\hline 0 & \\
\hline & 0 \\
\hline 1 & \\
\hline & 6
\end{array}
\end{aligned}
$$

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Problem 62

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&2 x-3 y=6\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline & 0 \\
\hline & 2 \\
\hline 2 &
\end{array}
\end{aligned}
$$

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Problem 63

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y+1=0\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline 1 & \\
\hline-3 & \\
\hline-1 &
\end{array}
\end{aligned}
$$

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Problem 64

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&x=3\\
&\begin{array}{c|c}
\boldsymbol{x} & \boldsymbol{y} \\
\hline & 0 \\
\hline & -2 \\
\hline & 3 \\
\hline & 1
\end{array}
\end{aligned}
$$

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Problem 65

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=\frac{1}{4} x+2\\
&\begin{array}{c|c}
\hline x & y \\
\hline 0 & \\
\hline-2 & \\
\hline 4 & \\
\hline-1 &
\end{array}
\end{aligned}
$$

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Problem 66

Complete the table of values for each equation, and plot the points.
$$
\begin{aligned}
&y=-\frac{5}{2} x+3\\
&\begin{array}{c|c}
x & y \\
\hline 0 & \\
\hline 4 & \\
\hline 2 & \\
\hline 1 & \\

\end{array}
\end{aligned}
$$

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Problem 67

For $y=\frac{2}{3} x-7$
a) find $y$ when $x=3, x=6,$ and $x=-3$. Write the results as ordered pairs.
b) find $y$ when $x=1, x=5,$ and $x=-2$. Write the results as ordered pairs.
c) why is it easier to find the $y$ -values in part a) than in part b)?

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Problem 68

Which ordered pair is a solution to every linear equation of the form $y=m x$, where $m$ is a real number?

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Problem 69

Fill in the blank with positive, negative, or zero.
The $x$ -coordinate of every point in quadrant III is _______.

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Problem 70

Fill in the blank with positive, negative, or zero.
The $y$ -coordinate of every point in quadrant I is ______.

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Problem 71

Fill in the blank with positive, negative, or zero.
The $x$ -coordinate of every point in quadrant II is ______.

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Problem 72

Fill in the blank with positive, negative, or zero.
The $y$ -coordinate of every point in quadrant II is _________.

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Problem 73

Fill in the blank with positive, negative, or zero.
The $x$ -coordinate of every point in quadrant I is ________.

Melissa Salvador

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Problem 74

Fill in the blank with positive, negative, or zero.
The $y$ -coordinate of every point in quadrant IV is ______.

Melissa Salvador

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Problem 75

Fill in the blank with positive, negative, or zero.
The $x$ -coordinate of every point on the $y$ -axis is ________.

Melissa Salvador

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Problem 76

Fill in the blank with positive, negative, or zero.
The $y$ -coordinate of every point on the $x$ -axis is _________.

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Problem 77

Solve Applied Problems Involving Ordered Pairs
The graph shows the number of people who visited Las Vegas from 2003 to 2008.
a) If a point on the graph is represented by the ordered pair $(x, y),$ then what do $x$ and $y$ represent?
b) What does the ordered pair (2004,37.4) represent in the context of this problem?
c) Approximately how many people went to Las Vegas in $2006 ?$
d) In which year were there approximately 38.6 million visitors?
e) Approximately how many more people visited Las Vegas in 2008 than in $2003 ?$
f) Represent the following with an ordered pair: During which year did Las Vegas have the most visitors, and how many visitors were there?

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Problem 78

Solve Applied Problems Involving Ordered Pairs
The graph shows the average amount of time people spent commuting to work in the Los Angeles metropolitan area from 2003 to 2007. (American Community Survey, U.S. Census)
a) If a point on the graph is represented by the ordered pair $(x, y),$ then what do $x$ and $y$ represent?
b) What does the ordered pair (2004,28.5) represent in the context of this problem?
c) Which year during this time period had the shortest commute? What was the approximate commute time?
d) When was the average commute 28.5 minutes?
e) Write an ordered pair to represent when the average commute time was 28.2 minutes.

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Problem 79

Solve Applied Problems Involving Ordered Pairs
The percentage of deadly highway crashes involving alcohol is given in the table.
$$
\begin{array}{c|c}
\text { Year } & \text { Percentage } \\
\hline 1985 & 52.9 \\
\hline 1990 & 50.6 \\
\hline 1995 & 42.4 \\
\hline 2000 & 41.4 \\
\hline 2005 & 40.0 \\
\hline
\end{array}
$$
a) Write the information as ordered pairs $(x, y)$ where $x$ represents the year and $y$ represents the percentage of accidents involving alcohol.
b) Label a coordinate system, choose an appropriate scale, and graph the ordered pairs.
c) Explain the meaning of the ordered pair (2000,41.4) in the context of the problem.

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Problem 80

Solve Applied Problems Involving Ordered Pairs
The average annual salary of a social worker is given in the table.
$$
\begin{array}{c|c}
\text { Year } & \text { Salary } \\
\hline 2005 & \$ 42,720 \\
\hline 2006 & \$ 44,950 \\
\hline 2007 & \$ 47,170 \\
\hline 2008 & \$ 48,180 \\
\hline
\end{array}
$$
a) Write the information as ordered pairs $(x, y)$ where $x$ represents the year and $y$ represents the average annual salary.
b) Label a coordinate system, choose an appropriate scale, and graph the ordered pairs.
c) Explain the meaning of the ordered pair (2007,47,170) in the context of the problem.

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Problem 81

Solve Applied Problems Involving Ordered Pairs
The amount of sales tax paid by consumers in Seattle in 2009 is given by $y=0.095 x$, where $x$ is the price of an item in dollars and $y$ is the amount of tax to be paid.
$$
\begin{array}{c|c}
\hline x & y \\
\hline 100.00 & \\
\hline 140.00 & \\
\hline 210.72 & \\
\hline 250.00 & \\
\hline
\end{array}
$$
a) Complete the table of values, and write the information as ordered pairs.
b) Label a coordinate system, choose an appropriate scale, and graph the ordered pairs.
c) Explain the meaning of the ordered pair (140.00,13.30) in the context of the problem.
d) How much tax would a customer pay if the cost of an item was $\$ 210.72 ?$
e) Look at the graph. Is there a pattern indicated by the points?
f) If a customer paid $\$ 19.00$ in sales tax, what was the cost of the item purchased?

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Problem 82

Solve Applied Problems Involving Ordered Pairs
Kyle is driving from Atlanta to Oklahoma City. His distance from Atlanta, $y$ (in miles), is given by $y=66 x$, where $x$ represents the number of hours driven.
a) Complete the table of values, and write the information as ordered pairs.
$$
\begin{array}{c|c}
\hline x & y \\
\hline 1 & \\
\hline 1.5 & \\
\hline 2 & \\
\hline 4.5 & \\
\hline
\end{array}
$$
b) Label a coordinate system, choose an appropriate scale, and graph the ordered pairs.
c) Explain the meaning of the ordered pair (4.5,297) in the context of the problem.
d) Look at the graph. Is there a pattern indicated by the points?
e) What does the 66 in $y=66 x$ represent?
f) How many hours of driving time will it take for Kyle to get to Oklahoma City if the distance between Atlanta and Oklahoma City is about 860 miles?

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Numerade Educator