Equations of Lines from
Graphs
Brenda Meery
Kaitlyn Spong
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Printed: January 28, 2015
AUTHORS
Brenda Meery
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Equations of Lines from Graphs
1
Equations of Lines from
Graphs
Here you’ll learn how to find the equation of a line from its graph.
Write the equation, in standard form, of the following graph:
Watch This
Khan Academy Slope and y-Intercept Intuition
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/58497
Khan Academy Slope 2
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/58498
1
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Guidance
You can determine the equation of a line from a graph by counting. Find the y-intercept (b) first and then a second
point on the line. Use the y-intercept and second point to determine the slope (m). Then, write the equation in slope
intercept form: y = mx + b.
The y-intercept of the graph is (0, –5). The slope of the line is 34 . The equation of the line in slope-intercept form is
3
y = x−5
4
.
If you cannot determine the y-intercept, you can algebraically determine the equation of a line by using the coordinates of two points on the graph. These two points can be used to calculate the slope of the line by counting and
then the y-intercept can then be determined algebraically.
To write the equation of a line in standard form, the value of the y-intercept is not needed. The slope can be
determined by counting. The value of the slope and the coordinates of one other point on the line are used in the
function y − y1 = m(x − x1 ). This equation is then set equal to 0 to write the equation in standard form.
Example A
Determine the equation of the following graph. Write the equation in slope-intercept form.
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Chapter 1. Equations of Lines from Graphs
Solution:
The y-intercept is (0, 4) so b = 4. The slope has a run of five units to the right and a rise of 2 units downward. The
slope of the line is − 25 . The equation of the line in slope-intercept form is y = mx + b so y = − 52 x + 4.
Example B
Determine the equation in slope-intercept form of the line shown on the following graph:
3
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Solution:
The y-intercept is not an exact point on this graph. The value of fractions on a Cartesian grid can only be estimated.
Therefore, the points (3, –1) and (9, –6) will be used to determine the slope of the line. The slope is − 65 . The slope
and one of the points will be used to algebraically calculate the y-intercept of the line.
y = mx + b
−5
−1 =
(3) + b
6
−5
−1 =
(3) + b
62
−5
−1 =
+b
2
5 −5 5
−1+ =
+ +b
2
2 2
5
−1 + = b
2
−2 5
+ =b
2
2
3
=b
2
The equation in slope-intercept form is
5
3
y = − x+
6
2
Example C
Determine the equation, in standard form, for the line on the following graph:
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Chapter 1. Equations of Lines from Graphs
Solution:
The y-intercept is not an exact point on this graph. Therefore, the points (4, 0) and (–1, –3) will be used to determine
the slope of the line. The slope is 35 . The slope and one of the points will be used to algebraically calculate the
equation of the line in standard form.
y − y1 = m(x − x1 )
3
y − 0 = (x − 4)
5
12
3
y = x−
5 5
3
12
5(y) = 5
x −5
5
5
3
12
5(y) = 5
x − 5
5
5
Use this formula to determine the equation in standard form.
3
x1 , y1
Fill in the value for m of and
4, 0
5
Multiply every term by 5.
Simplify and set the equation equal to zero.
5y = 3x − 12
5y−3x = 3x−3x − 12
5y−3x = −12
5y − 3x+12 = −12+12
5y − 3x + 12 = 0
−3x + 5y + 12 = 0
The coefficient of x cannot be a negative value.
3x − 5y − 12 = 0
The equation of the line in standard form is
3x − 5y − 12 = 0
.
Concept Problem Revisited
Write the equation, in standard form, of the following graph:
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The first step is to determine the slope of the line.
The slope of the line is 34 . The coordinates of one point on the line are (2, 5).
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Chapter 1. Equations of Lines from Graphs
y − y1 = m(x − x1 )
3
y − 5 = (x − 2)
4
3
6
y−5 = x−
4 4
3
6
4(y) − 4(5) = 4
x−4
4
4
3
6
x − 4
4(y) − 4(5) = 4
4
4
4y − 20 = 3x − 6
−3x + 4y − 20 = 3x − 3x − 6
−3x + 4y − 20 = −6
−3x + 4y − 20 + 6 = −6 + 6
−3x + 4y − 14 = 0
3x − 4y + 14 = 0
The equation of the line in standard form is
3x − 4y + 14 = 0
.
Vocabulary
Slope –Intercept Form
The slope-intercept form is one method for writing the equation of a line. The slope-intercept form is y =
mx + b where m refers to the slope and b identifies the y-intercept.
Standard Form
The standard form is another method for writing the equation of a line. The standard form is Ax + By +C = 0
where A is the coefficient of x, B is the coefficient of y and C is a constant.
Guided Practice
1. Write the equation, in slope-intercept form, of the following graph:
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2. Write the equation, in slope-intercept form, of the following graph:
3. Rewrite the equation of the line from #2 in standard form.
Answers:
1. The first step is to determine the coordinates of the y-intercept. The y-intercept is (0, –3) so b = −3. The second
step is to count to determine the value of the slope. Another point on the line is (7, 1) so the slope is 74 . The equation
of the line in slope-intercept form is
4
y = x−3
7
2. The y-intercept is not an exact point on the graph. Therefore begin by determining the slope of the line by counting
between two points on the line. The coordinates of two points on the line are (1, 0) and (6, –4). The slope is is − 45 .
The y-intercept of the line must be calculated by using the slope and one of the points on the line.
y = mx + b
−4
0=
(1) + b
5
−4
0=
+b
5
4 −4 4
0+ =
+ +b
5
5 5
4
=b
5
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Chapter 1. Equations of Lines from Graphs
The equation of the line in slope-intercept form is
4
4
y = − x+
5
5
3. To rewrite the equation in standard form, first multiply the equation by 5 to get rid of the fractions. Then, set the
equation equal to 0.
4
4
y = − x+
5
5
5y = −4x + 4
4x + 5y − 4 = 0
Explore More
For each of the following graphs, write the equation in slope-intercept form:
1. .
2. .
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3. .
4. .
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Chapter 1. Equations of Lines from Graphs
For each of the following graphs, write the equation in slope-intercept form:
5. .
6. .
7. .
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8. .
For each of the following graphs, write the equation standard form:
9. .
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Chapter 1. Equations of Lines from Graphs
10. .
11. .
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12. .
13.
14.
15.
16.
17.
14
Can you always find the equation of a line from its graph?
How do you find the equation of a vertical line? What about a horizontal line?
Rewrite the equation y = 14 x − 5 in standard form.
Rewrite the equation y = 23 x + 1 in standard form.
Rewrite the equation y = 13 x − 73 in standard form.