Parallel Lines

Learning Objectives

Introduction

Parallel linesLines that have the same slope and different `y`-intercepts. are two or more lines that never intersect. Examples of parallel lines are all around us, in the two sides of a page and in the shelves of a bookcase. When you see lines or structures that seem to run in the same direction, never cross one another, and are always the same distance apart, there’s a good chance that they are parallel.

In algebra, we use something more precise than appearance to recognize and create parallel lines. We use equations.

Recognizing Parallel Lines

Let’s look at two parallel lines on a graph.

A coordinate plane shows two parallel lines. Line A is labeled “y equals 3 x plus 4” and passes through (negative 1, 1). Line B is labeled “y equals 3 x minus 5” and passes through (2, 1).

Line A has an equation of `y=3x+4`. Line B has an equation of `y=3x-5`. Can you identify what these two lines have in common?

Try comparing the coefficients of `x` in each equation. They are both `3`. Because the equations are written in slope-intercept formA linear equation, written in the form `y = mx + b`, where `m` is the slope and `b` is the `y`-intercept. of `y=mx+b`, the coefficient of `x` is the slopeThe ratio of the vertical and horizontal changes between two points on a surface or a line. of the lines. Since both Line A and B have a slope of `3`, they have the same slope.

We’ve discovered a relationship that is true for all parallel lines—lines are parallel if they have the same slope. Not convinced? Look at the lines on this graph:

A coordinate plane shows two sets of parallel lines. The first set has two lines, labeled “y equals negative 2x plus 1” and “y equals negative 2x plus 6.” The second set has three lines, labeled “y equals one-fourth x plus 3,” “y equals one-fourth x minus 2,” and “y equals one-fourth x minus 3.”

The blue lines are parallel—they run in the same direction, keep the same distance apart, and never touch—and they have the same slope. All three red lines share a slope, and they, too, are parallel to one another. The red lines clearly are not parallel to the blue lines, and the red and blue lines have different slopes.

Creating Parallel Lines

Now that we’ve learned the relationship between parallel lines, let’s see if we can use that knowledge to create a parallel for a given line. Remember, parallel lines have the same slope but different y-interceptsThe point where the graph of a linear equation intersects the `y`-axis (`0`,`y`)..

To draw a line parallel to a line on a graph, we can lay a second line directly on the first, and then slide it off along one of the axes. That will create a parallel line. Try it on the graph below. Use the green slider bar labeled `m` (for slope) to swivel the red line on top of the blue one. Then use the other green slider labeled `b` (for `y`-intercept) to shift the red line off the blue line while maintaining the same slope. That will give you two parallel lines.

Notice that it is possible to create an infinite number of parallel lines on the graph.

To create a parallel line from an equation, we start by identifying the slope of the original line. Then we write a second equation, repeating the slope but changing the `y`-intercept. Once again, because there are an infinite number of possible `y`-intercepts, there are an infinite number of equations we could create.

A line has the equation `y=2/3x+7`. Which of the following lines is parallel to it?

 

A) `y=1/3x+7`

 

B) `y=2/3x-7`

 

C) `y=2/3x+7`

 

D) `y=-2/3x+2`

 

 

Summary

Parallel lines are lines that have the same slope. These lines never intersect and always maintain the same distance apart. To create a parallel line from an equation, we simply change the `y`-intercept.