## Parallel Lines in the Coordinate Plane

## Introduction

Train tracks can be found all throughout the world. In order for trains to be able to run on the tracks, the two rails must remain the same exact distance apart for their entire length. The rails on a straight track can be described as *parallel*. They look like lines that point in the same direction yet never touch.

You should already be familiar with the equations of lines on the coordinate plane. Now suppose that you can superimpose a pair of parallel lines onto the coordinate plane. This lesson will examine those parallel lines and the similarities and differences in the equations that form parallel lines.

## Lesson

There are two parts to this lesson.

- Determining whether two existing lines are parallel (or not)
- Constructing a line that is parallel to a known line

**Determining whether two existing lines are parallel (or not)**

It is pretty easy to determine whether lines are parallel when they are on a graph. Looking at the graph to the right, line #1 and line #2 are parallel. Move your cursor over the graph to reveal line #3. Although Line #3 points in a similar direction, it is not parallel to the other two lines.

Focus on line #1 compared to line #2. The equations of these two lines are shown on the diagram to the left. While the lines have different y-intercepts, their slopes are the same. This is true of all parallel lines.

Now take a look at line #1 compared to #3. The equations of these two lines are shown on the diagram to the left. The key to determining whether they are parallel is determining whether their slopes are the same. Since their slopes are not the same, then the lines are not parallel.

**Example 1:** Determine if each pair of lines is parallel.

y = 3x + 2 and y = 3x + 6

Parallel: These lines have the same slope. Therefore they are parallel.

3y = -6x + 21 and y = -2x – 6

Parallel: The first equation can be changed to y = -2x + 7, so both lines have a slope of -2 and are parallel.

5x + 5y = 40 and 3y = 3x + 9

Not parallel: The first equation can be simplified to y = -x + 8. The second equation can be simplified to y = x + 3. The slopes are different (-1 and 1) so the lines are not parallel.

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**Constructing a line that is parallel to a given line**

Every point on the coordinate plane can be a part of an infinite number of lines. Consider point A on the coordinate plane to the right. There are a number of different lines that contain the point A. However, there is only one way to draw a line through point A that is parallel to line B.

Every two parallel lines have the same slope. This information can be used to find the equation of that one line that goes through point A and is parallel to line B. Example 2 shows how to find this line.

**Example 2****:** Find the equation of the line that goes through the point (4, 6) and is parallel to the line y =x + 3.

Hint: Since the lines are parallel, the slopes must be the same.

Here is one more example of finding the equation of a missing parallel line.

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**Example 3****:** Find the equation of the line that goes through the point (-2, 5) and is parallel to the line y = 2x – 1.

## Review

1) Find the equation of the line that goes through the point (2, 5) and is parallel to the line y = 5x + 1.

2) Find the equation of the line that goes through the point (4, -4) and is parallel to the line 6x + 3y = 12.

Determine whether the lines are parallel

3) y = 2x + 4 and y = 2x + 15

4) y + 8 = -3x and y = 2 – 3x

5) 2x + 4y = 10 and y = 2x

6) 3y = 9x + 27 and 2y – 6x = 6

Scroll down for answers…

**Answers****:**** **

1) The parallel line has an equation of y = 5x – 10.

2) The paralle line has an equation of y = -2x.

Determine whether the lines are parallel

3) Yes. Both lines have slopes of 2.

4) Yes. Both lines have slopes of -3.

5) No. Their slopes are different. (– ½ and 2)

6) Yes. Both lines have slopes of 3.

**Related Links: **

For more information on lines and their properties, try one of the links below.

**Resource Pages**

- Graphing Linear Equations (Intro) Resource Page
- Graphing Linear Equations (Advanced) Resource Page

**Related Lessons**

- The Slope-Intercept Form of an Equation

- Find the Equation (Given 1 point and slope)
- Find the Equation (Given 2 points)
- Perpendiciular Lines in the Coordinate Plane

Looking for something else? Try the general math or algebra lessons.