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2.4: Finding the Equation of Parallel Lines

Difficulty Level: At Grade Created by: CK-12
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Pablo is researching different types of exercise to see which gives the best calorie burn. He found that an indoor cycling class can burn up to 150 calories every 10 minutes. The elliptical machine burns 270 calories every 30 minutes. Do these two forms of exercise burn the same amount of calories in an hour?

Guidance

When two lines are parallel, they have the same slope and never intersect. So, if a given line has a slope of -2, then any line that is parallel to that line will also have a slope of -2, but it will have a different yintercept.

Example A

Find the equation of the line that is parallel to y=23x5 and passes through (-12, 1).

Solution: We know that the slopes will be the same; however we need to find the yintercept for this new line. Use the point you were given, (-12, 1) and plug it in for x and y to solve for b.

y119=23x+b=23(12)+b=8+b=b

The equation of the parallel line is y=23x+9.

Example B

Write the equation of the line that passes through (4, -7) and is parallel to y=2.

Solution: The line y=2 does not have an xterm, meaning it has no slope. This is a horizontal line. Therefore, to find the horizontal line that passes through (4, -7), we only need the ycoordinate. The line would be y=7.

The same would be true for vertical lines, but all vertical line equations are in the form x=a. The xcoordinate of a given point would be what is needed to determine the equation of the parallel vertical line.

Example C

Write the equation of the line that passes through (6, -10) and is parallel to the line that passes through (4, -6) and (3, -4).

Solution: First, we need to find the slope of the line that our line will be parallel to. Use the points (4, -6) and (3, -4) to find the slope.

m=4(6)34=21=2

This is the slope of our given line as well as the parallel line. Use the point (6, -10) to find the yintercept of the line that we are trying to find the equation for.

10102=2(6)+b=12+b=b

The equation of the line is y=2x+2.

Intro Problem Revisit The equation for indoor cycling is y=15010x or y=15x. For the elliptical, the equation would be y=27030x or y=9x. The two lines are not parallel, so the two exercises do not burn an equal number of calories. After 60 minutes, indoor cycling will burn 900 calories and the elliptical will burn 540 calories.

Guided Practice

1. Find the equation of the line that is parallel to x2y=8 and passes through (4, -3).

2. Find the equation of the line that is parallel to x=9 and passes through (-1, 3).

3. Find the equation of the line that passes through (-5, 2) and is parallel to the line that passes through (6, -1) and (1, 3).

Answers

1. First, we need to change this line from standard form to slope-intercept form.

x2y2yy=8=x+8Now, we know the slope is 12. Let's find the new y intercept.=12x4

335=12(4)+b=2+b=b

The equation of the parallel line is y=12x5 or x2y=10.

2. x=9 is a vertical line that passes through the xaxis at 9. Therefore, we only need to xcoordinate of the point to determine the equation of the parallel vertical line. The parallel line through (-1, 3) would be x=1.

3. First, find the slope between (6, -1) and (1, 3).

m=1361=45=45

This will also be the slope of the parallel line. Use this slope with the given point, (-5, 2).

221=45(5)+b=1+b=b

The equation of the parallel line is y=45x+1.

Vocabulary

Parallel
When two or more lines are in the same plane and never intersect. These lines will always have the same slope.

Practice

Find the equation of the line, given the following information. You may leave your answer in slope-intercept form.

  1. Passes through (4, 7) and is parallel to xy=5.
  2. Passes through (-6, -2) and is parallel to y=4.
  3. Passes through (-3, 5) and is parallel to y=13x1.
  4. Passes through (1, -9) and is parallel to x=8.
  5. Passes through the yintercept of 2x3y=6 and parallel to x4y=10.
  6. Passes through (-12, 4) and is parallel to y=3x+5.
  7. Passes through the xintercept of 2x3y=6 and parallel to x+4y=3.
  8. Passes through (7, -8) and is parallel to 2x+5y=14.
  9. Passes through (1, 3) and is parallel to the line that passes through (-6, 2) and (-4, 6).
  10. Passes through (-18, -10) and is parallel to the line that passes through (-2, 2) and (-8, 1).
  11. Passes through (-4, -1) and is parallel to the line that passes through (15, 7) and (-1, -1).

Are the pairs of lines parallel? Briefly explain how you know.

  1. x2y=4 and 5x+10y=16
  2. \begin{align*}3x + 4y = -8\end{align*} and \begin{align*}6x + 12y = -1\end{align*}
  3. \begin{align*}5x - 5y = 20\end{align*} and \begin{align*}x + y = 7\end{align*}
  4. \begin{align*}8x - 12y = 36\end{align*} and \begin{align*}10x - 15y = -15\end{align*}

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Vocabulary

Parallel

Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.

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Date Created:
Mar 12, 2013
Last Modified:
Jun 07, 2016
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