Finding Parallel Lines

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  • 0:05 Parallel Line Review
  • 1:39 Finding Parallel Lines
  • 3:22 Example
  • 4:27 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

If you're given a line, can you provide the equation of a second line parallel to the original? In this lesson, you'll learn to do just that, whether you just need a parallel line or whether you need one that goes through a particular point.

Parallel Line Review

In this lesson, you'll learn how to find the equation of a line parallel to a given line. But first, we'll review some important facts about parallel lines that will help you figure out the math.

Parallel lines are lines with the same slope but different y-intercepts. They're going in the same direction, but they never touch each other; if two lines intercept at any point, then they're not parallel.

To find out whether two lines are parallel, you can put the equation of each line into a slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. Then simply compare the values of m and b. If m is the same but b is different, the lines are parallel.

For example, take the two lines:

2x - 1 = y
y + 3x = 4

If you put both lines into the form y = mx + b, you get y = 2x -1 and y = -3x + 4. 2 isn't the same as -3, so the slopes of these lines are not the same. That means they aren't parallel.

A pair of parallel lines would be 2x - 1 = y and y - 2x = 4. If you put both of these lines into y = mx + b form, you get y = 2x - 1 and y = 2x + 4. These lines have the same slope but different y-intercepts, so they are parallel.

Finding Parallel Lines

So far, so good. But what if you get a question like this?

Find the equation of a line parallel to the line y - 2x = 4

First, you have to put the given line into y = mx + b form, which would give you y = 2x + 4. Then, you have your choice of answers because any line with a slope of 2 and a y-intercept that isn't 4 would be parallel to this line. You could pick y = 2x - 3, or y = 2x + 7, or any other line you like.

But what if we make things more complicated? What about this question:

Find the equation of a line parallel to the line y - 2x = 4 that passes through the point (1, -1)

Now you can't just pick any line. Instead, you have to solve for the equation of this particular line. Once again, we'll start by writing the line we have in y = mx + b form: y = 2x + 4. We know we're looking for a line with a slope of 2 because it has to be parallel to our original line. The new line also has to pass through the point (1, -1).

We can plug that point into the equation of the line: (-1) = 2*(1) + b. Now we'll do the math. -1 = 2 + b. Subtract 2 from both sides and you get -3 = b. We've now solved for the y-intercept of our line. We know the slope is 2 and the y-intercept is -3, so we know the line we're looking for is the line y = 2x - 3.

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