Parallel Lines in the
Coordinate Plane
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: November 21, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Parallel Lines in the Coordinate Plane
1
Parallel Lines in the
Coordinate Plane
Here you’ll learn that parallel lines have the same slope. You’ll then apply this fact to determine if two lines are
parallel and to find what their equations are.
What if you were given two parallel lines in the coordinate plane? What could you say about their slopes? After
completing this Concept, you’ll be able to answer this question. You’ll also find the equations of parallel lines and
determine if two lines are parallel based on their slopes.
Watch This
MEDIA
Click image to the left for more content.
CK-12 Parallel Linesin the Coordinate Plane
Guidance
Parallel lines are two lines that never intersect. In the coordinate plane, that would look like this:
If we take a closer look at these two lines, the slopes are both 23 .
This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different
y−intercepts.
Example A
Find the equation of the line that is parallel to y = − 13 x + 4 and passes through (9, -5).
Recall that the equation of a line is y = mx + b, where m is the slope and b is the y−intercept. We know that parallel
lines have the same slope, so the line will have a slope of − 31 . Now, we need to find the y−intercept. Plug in 9 for x
and -5 for y to solve for the new y−intercept (b).
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1
−5 = − (9) + b
3
−5 = −3 + b
−2 = b
The equation of parallel line is y = − 13 x − 2.
Example B
Find the equation of the lines below and determine if they are parallel.
The top line has a y−intercept of 1. From there, use “rise over run” to find the slope. From the y−intercept, if you
go up 1 and over 2, you hit the line again, m = 12 . The equation is y = 12 x + 1.
For the second line, the y−intercept is -3. The “rise” is 1 and the “run” is 2 making the slope 12 . The equation of this
line is y = 12 x − 3.
The lines are parallel because they have the same slope.
Example C
Find the equation of the line that is parallel to the line through the point marked with a blue dot.
First, notice that the equation of the line is y = 2x + 6 and the point is (2, -2). The parallel would have the same slope
and pass through (2, -2).
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Concept 1. Parallel Lines in the Coordinate Plane
y = 2x + b
−2 = 2(2) + b
−2 = 4 + b
−6 = b
The equation of the parallel line is y = 2x + −6.
MEDIA
Click image to the left for more content.
CK-12 Parallel Linesin the Coordinate Plane
Guided Practice
1. Find the equation of the line that is parallel to y = 41 x + 3 and passes through (8, -7).
2. Are the lines y = 2x + 1 and y = 2x + 5 parallel?
3. Are the lines 3x + 4y = 7 and y = 34 x + 1 parallel?
Answers:
1. We know that parallel lines have the same slope, so the line will have a slope of 14 . Now, we need to find the
y−intercept. Plug in 8 for x and -7 for y to solve for the new y−intercept (b).
1
−7 = (8) + b
4
−7 = 2 + b
−9 = b
The equation of the parallel line is y = 14 x − 9.
2. Both equations are already in slope-intercept form and their slopes are both 2 so yes, the lines are parallel.
3. First we need to rewrite the first equation in slope-intercept form.
3x + 4y = 7
4y = −3x + 7
3
7
y = − x+
4
4
.
The slope of this line is − 34 while the slope of the other line is 34 . Because the slopes are different the lines are not
parallel.
3
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Practice
Determine if each pair of lines are parallel. Then, graph each pair on the same set of axes.
1.
2.
3.
4.
y = 4x − 2 and y = 4x + 5
y = −x + 5 and y = x + 1
5x + 2y = −4 and 5x + 2y = 8
x + y = 6 and 4x + 4y = −16
Determine the equation of the line that is parallel to the given line, through the given point.
5.
6.
7.
8.
y = −5x + 1; (−2, 3)
y = 32 x − 2; (9, 1)
x − 4y = 12; (−16, −2)
3x + 2y = 10; (8, −11)
Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.
9.
For the line and point below, find a parallel line, through the given point.
10.
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Concept 1. Parallel Lines in the Coordinate Plane
11.
12.
13.
5