Graphing Two-Variable
Equations
Connections
Have you ever . . .
• Found a graph easier to read than text?
• Found the coordinates of a town on a map?
• Played Battleship™?
Graphing is a visual way to present information,
whether it is a graph that shows changing government spending or a map showing that Georgetown
is at C-7. The game Battleship by Hasbro® uses the
concept of graphing, too. Each ship is located at a
set of points on the graph.
A coordinate graph has two axes, lines that represent the things whose relationship is being
compared. The lines intersect at a right angle to make a coordinate plane, or coordinate
axes. A line can be described in slope-intercept form with its slope, m, and its y-intercept, b:
y = mx + b
This equation gives you a lot of information about a line.
• If the slope is positive, the line climbs from left to right.
• If the slope is negative, the line falls from left to right.
• The larger the absolute value of the slope, the steeper the line.
• The equation shows you the y-intercept, where x = 0.
To graph an equation on a coordinate plane, you can find three points that make the equation true, and draw them on the coordinate axes. The line going through these points is the
graph of the equation.
141
Essential Math Skills
Learn
It!
One, Two, Three, Graph!
Graphing is a way to represent data visually. An equation like y = 7x + 3 tells you
a lot about the relationship between x and y. But the equation can’t instantly
show you what y is if x is 23. A graph can. You can draw a graph in three steps:
• Find three points that satisfy the equation (make the equation true).
• Draw the points on the coordinate plane.
• Draw a straight line through the three points.
The cost to rent a copier for a day is $50 plus $0.05 per copy. To help her employees understand the company’s expenses, Chanry posts a graph of the total costs
per copy above the copier. What does the graph look like?
Find the Equation
Use the slope-intercept form y = mx + b or the point-slope form y - y1 = m(x - x1) to find an
equation for the graph. The slope (m) is the rate of change, and the y-intercept (b) is a constant value. Use the point-slope form if you know a point on the graph.
?
1. What is the equation relating daily cost for the copier, y, to number of copies, x?
Use the slope-intercept form, since you have a rate of change and a constant value. The rate
per copy is 0.05, so it is the slope (m). The y-intercept (b) is the amount of y if x = 0. In this
case, it is the cost per day if the office makes zero copies ($50). The equation is:
y = 0.05x + 50
One, Two, Three: A Table of Three Points
You could graph the line with the slope and y-intercept, but if you graph the line using three
points, you can check for errors. A line is defined by just two points. By using the third point,
you have a checkpoint to confirm that your line is correct.
Pick three easy-to-use values of x, such as 0, 1, and 2 or 0, 100, and 200. For each x, find y
using the equation of the line. Put the three ordered pairs (x, y) into a table.
?
142
2. Fill out the table with three ordered pairs that satisfy the equation.
x
y
Graphing Two-Variable Equations
Substituting 0, 100 and 200 into the equation, you get the following:
Slope-Intercept
x
y
y = 0.05(0) + 50 = 50
0
50
y = 0.05(100) + 50 = 55
100
55
y = 0.05(200) + 50 = 60
200
60
Math Tip
Write your equation
in slope-intercept
form to easily find
ordered pairs that
satisfy it.
Graph: Plot the Points and Draw the Line
Find each point on the coordinate plane and make a dot. The x coordinate is always the horizontal axis, and the y coordinate is always the vertical axis. When all three points are on the
graph, one straight line should go through them. If not, one of the points is wrong. Draw the
line, and check its meaning against the original problem. Where should it start and end?
?
3. Plot the three points on the graph and draw the line representing the cost of copies.
y
200
Build Your
Math Skills
150
100
The variable on
the y-axis is the
dependent variable,
the variable that
you are trying to
find based on the
value of x.
50
0
–200 –150 –100 –50
x
50
100
150
200
–50
–100
The variable on
the x-axis is the
independent
variable, the
variable that makes
a change in the
value of y.
–150
–200
The completed line should
look like this graph. Notice
that the line should start at
the y-intercept, since you
can’t make a negative number of copies.
What does the graph tell you
about the cost of copies?
y
200
150
100
50
0
x
50
100
150
200
For example, if you
want to know how
the price of bread
changes based on
the price of flour,
graph bread as y
and flour as x.
143
Essential Math Skills
e
ic
Pract
It!
Graph the following linear equations.
1. Graph the equation y = 7x + 3.
a. Complete the table of points.
b. Graph the equation.
y
x
y
10
9
8
7
6
5
4
3
2
1
0
1
-1
c. How you would graph the equation
without finding three points?
Compare the two methods.
–10–9 –8 –7 –6 –5 –4 –3 –2 –1–1
0
x
1 2 3 4 5 6 7 8 9 10
–2
–3
–4
–5
–6
–7
–8
–9
–10
2.Graph 2y + 6 = x + 10.
a. Make a table of points.
b. Graph the equation.
y
x
144
y
c. What does the line tell you about
the relationship of x and y?
10
9
8
7
6
5
4
3
2
1
–10–9 –8 –7 –6 –5 –4 –3 –2 –1–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
0
x
1 2 3 4 5 6 7 8 9 10
Graphing Two-Variable Equations
3. A car rental agency charges $25 per day plus $0.08 for every mile driven.
a. Write an equation for the cost c of renting the car for one day and driving m miles.
b. Make a table of points.
m
c
c. Graph the equation.
c
100
90
80
70
60
d. What benefits are there to looking
at this relationship on a graph?
50
40
30
20
10
m
0
10 20 30 40 50 60 70 80 90 100
4. Norma’s job pays $1,200 per month plus $45 for each new client she attracts.
a. Write an equation for Norma’s monthly wage w based on her new clients a.
b. Make a table of points.
a
w
c. Graph the equation.
w
3000
2700
2400
2100
1800
d. The scales of the x-axis (a) and
y-axis (w) are different. How does
this affect the graph?
1500
1200
900
600
300
0
a
3
6
9
12 15 18 21 24 27 30
145
Essential Math Skills
5. Atsidi and Jorge are working on graphing the line -3d = t + 24. Jorge drew this graph.
d
a. How could Atsidi know immediately
the graph was wrong?
10
8
6
4
b. Identify the mistake. What advice
would you give Jorge to avoid this
mistake in the future?
2
t
0
–12 –10 –8 –6
–4 –2
2
4
6
8
10 12
–2
–4
–6
–8
–10
c.
Draw the correct line on the graph.
–12
–14
6. Kalil felt very confident of his graphing skills and only used two points to graph
his homework. Sometimes this worked out for him, but when he was graphing
1.1y - 4.4 = -6.05x, he got the answer wrong and didn’t have the check point to alert
him of the error.
y
a. Identify Kalil’s mistake. What advice
would you give Kalil to avoid this
mistake in the future?
30
25
20
15
10
5
x
0
–30 –25 –20 –15 –10 –5
5
–5
–10
b. Draw the correct line on the graph.
–15
–20
–25
–30
146
10 15 20 25 30
Graphing Two-Variable Equations
Check Your Skills
Check your graphing skills by answering the following questions.
5
1.Graph y = 3x + 2 .
Math Tip
Use parentheses
when substituting a
negative value of x
to avoid errors.
y
20
15
10
5
x
0
–20 –15 –10 –5
5
10 15 20
–5
–10
–15
–20
2. A local cell phone tower has developed this equation to model the growth over its user
base over the last five years: p = 85.6x + 3000.
a. Complete the table of values.
x
b. Graph the equation.
p
p
0
3500
1
3400
5
3300
3200
3100
3000
3.
x
0
1
2
3
4
5
The cost of manufacturing one type of calculator is $500 for the initial equipment
plus $6.20 per calculator for labor and materials. Which point is not on the graph
of the cost y of manufacturing x calculators?
a. (5, 531)
b. (10, 562)
c. (20, 724)
d. (25, 676)
147
Essential Math Skills
4. What do the graphs of y = 14x - 7 and y = -14x + 7 have in common?
a. The slope
b.The x-intercept
c.The y-intercept
d. The point (0.4, 1.4)
5. A cave near your house has a stalactite that grows 0.2 inches per year. When you
started measuring the stalactite, it was 4.5 inches long. Which equation represents the
length l of the stalactite over t years from the date you started measuring it?
a. l = 0.2t + 4.5
b. l = -0.2t + 4.5
c. l = 4.5t + 0.2
d. l = 4.5t - 0.2
6. Circle the graph that shows the equation 4y + 7 = 3x - 2.
10
8
6
4
2
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
y
2 4 6 8 10
x
10
8
6
4
2
y
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
2 4 6 8 10
x
10
8
6
4
2
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
y
2 4 6 8 10
x
10
8
6
4
2
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
y
2 4 6 8 10
x
7. Your cell phone cost $425. It depreciates (loses value) linearly and loses $280 in value
over three years. Draw a graph that models this depreciation.
y
500
400
Remember
the Concept
300
200
100
x
0
148
1
2
3
4
5
One, Two, Three,
Graph!
Three points will define
and check your line.
Answers and Explanations
Graphing Two-Variable
Equations
2b.
y
page 141
10
9
8
7
6
5
4
3
2
1
One, Two, Three, Graph!
Practice It!
pages 144–146
1a.
x
y
0
3
1
10
–10–9 –8 –7 –6 –5 –4 –3 –2 –1–1
-1
-4
–2
–3
–4
–5
–6
–7
–8
–9
–10
1b.
y
10
9
8
7
6
5
4
3
2
1
–10–9 –8 –7 –6 –5 –4 –3 –2 –1–1
0
x
1 2 3 4 5 6 7 8 9 10
2c.Answers will vary. The graph tells you that x increases
more quickly than y.
0
x
1 2 3 4 5 6 7 8 9 10
–2
–3
–4
–5
–6
–7
–8
–9
–10
3a.c = 0.08m + 25
3b.
m
c
0
25
10
25.8
50
29
3c.
c
1c.You could graph the equation by using the y-intercept
(0, 3) and the slope (seven). Find the point (0, 3)
and graph a line that slopes up seven and right one.
Comparisons will vary.
2a.
100
90
80
70
60
x
y
0
2
2
3
30
-2
1
20
50
40
10
0
m
10 20 30 40 50 60 70 80 90 100
3d.Answers will vary. Looking at the relationship on a
graph visually shows the increase per mile and allows
you to easily check cost based on mileage.
4a.w = 45a + 1200
i
Essential Math Skills
4b.
4c.
a
w
0
1200
10
1650
20
2100
You can rewrite the equation in terms of d.
1
d=- t-8
3
t
d
0
-8
-3
-7
3
-9
w
3000
6a.You can rewrite the equation in slope-intercept form.
2700
1.1y - 4.4 = -6.05x
2400
y = -5.5x + 4
2100
The slope is negative, but the graph shows a positive
(right-slanting) slope. You can identify that Kalil made
a mistake with negative numbers. You might advise
him to plot three points next time.
1800
1500
1200
900
6b.
600
y
300
0
30
a
3
6
9
25
12 15 18 21 24 27 30
20
4d.The different scales allow you to see how large numbers relate to small numbers, but the actual slope of
the line is distorted.
5a.Because there are no exponents on the variables,
3d = t + 24 is a linear equation. Its graph should be a
straight line.
5b.Jorge made an error with negatives when he found
the point based on t = -3. When t = -3, d = -7. You
might advise Jorge to put parentheses around negative numbers so he doesn’t forget the negative signs.
5c.
d
8
6
4
2
t
0
–4 –2
2
–2
–4
–6
–8
–10
–12
–14
ii
10
5
4
6
8
10 12
x
0
–30 –25 –20 –15 –10 –5
5
–5
–10
–15
–20
–25
–30
10
–12 –10 –8 –6
15
10 15 20 25 30
Answers and Explanations
Check Your Skills
pages 147–148
1.
10
8
6
4
2
y
20
15
10
5
x
0
–20 –15 –10 –5
6.
5
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
y
2 4 6 8 10
x
10
8
6
4
2
y
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
2 4 6 8 10
x
10
8
6
4
2
–10 –8 –6 –4 –2
–2
–4
–6
–8
–10
y
2 4 6 8 10
10 15 20
–5
4y + 7 = 3x - 2
–10
4y = 3x - 9
–15
y =
–20
3
9
4x 4
7.
y
2a.
x
p
500
0
3000
400
1
3085.6
300
5
3428
200
2b.
100
p
x
0
3500
1
2
3
4
5
280
y = - 3 x + 425
3400
3300
3200
3100
3000
x
0
1
2
3
4
5
3.
d. (25, 676)
y = 6.2x + 500
y = 6.2(25) + 500 = 655 ! 676
4.
b. The x-intercept
1
When y = 0, x = 2 for both lines.
5.
a. l = 0.2t + 4.5
iii
x
–1