Ulearngo

Using Slopes to Identify Parallel Lines

Using Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See the figure below.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).

Verify that both lines have the same slope, \(m=\frac{2}{5}\), and different y-intercepts.

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel. See the figure below.

The figure shows two vertical lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (2,1) and (2,5). The other line goes through the points (5, negative 4) and (5,0).

Vertical lines with diferent x-intercepts are parallel.

Parallel Lines

Parallel lines are lines in the same plane that do not intersect.

  • Parallel lines have the same slope and different y-intercepts.
  • If \({m}_{1}\) and \({m}_{2}\) are the slopes of two parallel lines then\({m}_{1}={m}_{2}\).
  • Parallel vertical lines have different x-intercepts.

Let’s graph the equations \(y=-2x+3\) and \(2x+y=-1\) on the same grid. The first equation is already in slope–intercept form: \(y=-2x+3\). We solve the second equation for \(y\):

\(\begin{array}{ccc}\hfill 2x+y& =\hfill & -1\hfill \\ \hfill y& =\hfill & -2x-1\hfill \end{array}\)

Graph the lines.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 4, 7) and (3, negative 7). The other line goes through the points (negative 2, 7) and (5, negative 7).

Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?

\(\begin{array}{cccccccccc}\hfill y& =\hfill & mx+b\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill y& =\hfill & mx+b\hfill \\ \hfill y& =\hfill & -2x+3\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill y& =\hfill & -2x-1\hfill \\ \hfill m& =\hfill & -2\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill m& =\hfill & -2\hfill \\ \hfill b& =\hfill & 3,\text{(0, 3)}\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill b& =\hfill & -1,\text{(0, −1)}\hfill \end{array}\)

The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.

Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

Example

Use slopes and y-intercepts to determine if the lines \(3x-2y=6\) and \(y=\frac{3}{2}x+1\) are parallel.

Solution

\(\begin{array}{cccccccccc}\begin{array}{}\text{Solve the first equation for}\phantom{\rule{0.2em}{0ex}}y.\hfill \\ \text{The equation is now in slope–intercept form.}\hfill \\ \end{array}\hfill & & & \phantom{\rule{1.5em}{0ex}}\begin{array}{ccc}\hfill 3x-2y& =\hfill & 6\hfill \\ \hfill -2y& =\hfill & -3x+6\hfill \\ \hfill \frac{-2y}{-2}& =\hfill & \frac{-3x+6}{-2}\hfill \\ \hfill y& =\hfill & \frac{3}{2}x-3\hfill \end{array}\hfill & & & \hfill \begin{array}{c}\hfill \text{and}\hfill \\ \end{array}\hfill & & & \begin{array}{c}\phantom{\rule{0.2em}{0ex}}y=\frac{3}{2}x+1\hfill \\ \end{array}\hfill \\ \begin{array}{c}\text{The equation of the second line is already}\hfill \\ \text{in slope–intercept form.}\hfill \\ \text{Identify the slope and}\phantom{\rule{0.2em}{0ex}}y\text{-intercept of both lines.}\hfill \\ \end{array}\hfill & & & \phantom{\rule{3.7em}{0ex}}\begin{array}{}\\ \hfill y& =\hfill & \frac{3}{2}x-3\hfill \\ \hfill y& =\hfill & mx+b\hfill \\ \hfill m& =\hfill & \frac{3}{2}\hfill \end{array}\hfill & & & & & & \begin{array}{c}\phantom{\rule{0.2em}{0ex}}y=\frac{3}{2}x+1\hfill \\ \phantom{\rule{0.2em}{0ex}}y=\frac{3}{2}x+1\hfill \\ \phantom{\rule{0.2em}{0ex}}y=mx+b\hfill \\ m=\frac{3}{2}\hfill \end{array}\hfill \\ & & & \phantom{\rule{2.5em}{0ex}}y\text{-intercept is (0, −3)}\hfill & & & & & & y\text{-intercept is (0, 1)}\hfill \end{array}\)

The lines have the same slope and different y-intercepts and so they are parallel. You may want to graph the lines to confirm whether they are parallel.

Example

Use slopes and y-intercepts to determine if the lines \(y=-4\) and \(y=3\) are parallel.

Solution

\(\begin{array}{cccc}\begin{array}{}\text{Write each equation in slope–intercept form.}\hfill \\ \text{Since there is no}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{term we write}\phantom{\rule{0.2em}{0ex}}0x.\hfill \\ \text{Identify the slope and}\phantom{\rule{0.2em}{0ex}}y\text{-intercept of both lines.}\hfill \end{array}\hfill & & & \phantom{\rule{2em}{0ex}}\begin{array}{ccccccc}\phantom{\rule{0.3em}{0ex}}y=-4\hfill & & & \text{and}\hfill & & & \phantom{\rule{0.3em}{0ex}}y=3\hfill \\ \phantom{\rule{0.3em}{0ex}}y=0x-4\hfill & & & & & & \phantom{\rule{0.3em}{0ex}}y=0x+3\hfill \\ \phantom{\rule{0.3em}{0ex}}y=0x-4\hfill & & & & & & \phantom{\rule{0.3em}{0ex}}y=0x+3\hfill \\ \phantom{\rule{0.3em}{0ex}}y=mx+b\hfill & & & & & & \phantom{\rule{0.3em}{0ex}}y=mx+b\hfill \\ m=0\hfill & & & & & & m=0\hfill \\ y\text{-intercept is (0, 4)}\hfill & & & & & & y\text{-intercept is (0, 3)}\hfill \end{array}\hfill \end{array}\)

The lines have the same slope and different y-intercepts and so they are parallel.

There is another way you can look at this example. If you recognize right away from the equations that these are horizontal lines, you know their slopes are both 0. Since the horizontal lines cross the y-axis at \(y=-4\) and at \(y=3\), we know the y-intercepts are \(\left(0,-4\right)\) and \(\left(0,3\right)\). The lines have the same slope and different y-intercepts and so they are parallel.

Example

Use slopes and y-intercepts to determine if the lines \(x=-2\) and \(x=-5\) are parallel.

Solution

\(x=-2\phantom{\rule{0.4em}{0ex}}\text{and}\phantom{\rule{0.4em}{0ex}}x=-5\)

Since there is no \(y\), the equations cannot be put in slope–intercept form. But we recognize them as equations of vertical lines. Their x-intercepts are \(-2\) and \(-5\). Since their x-intercepts are different, the vertical lines are parallel.

Example

Use slopes and y-intercepts to determine if the lines \(y=2x-3\) and \(-6x+3y=-9\) are parallel. You may want to graph these lines, too, to see what they look like.

Solution

\(\begin{array}{cccccccc}\begin{array}{}\text{The first equation is already in slope–intercept form.}\hfill \\ \text{Solve the second equation for}\phantom{\rule{0.2em}{0ex}}y.\hfill \\ \end{array}\hfill & & & \phantom{\rule{1.5em}{0ex}}\begin{array}{ccc}\hfill y& =\hfill & 2x-3\hfill \\ \hfill y& =\hfill & 2x-3\hfill \\ \hfill -6x+3y& =\hfill & -9\hfill \\ \hfill 3y& =\hfill & 6x-9\hfill \\ \hfill \frac{3y}{3}& =\hfill & \frac{6x-9}{3}\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill \text{and}\hfill \\ \end{array}\hfill & & & \begin{array}{c}-6x+3y=-9\hfill \\ \end{array}\hfill \\ \begin{array}{}\\ \text{The second equation is now in}\hfill \\ \text{slope–intercept form.}\hfill \\ \text{Identify the slope and}\phantom{\rule{0.2em}{0ex}}y\text{-intercept of both lines.}\hfill \\ \end{array}\hfill & & & \phantom{\rule{4.5em}{0ex}}\begin{array}{ccc}\hfill y& =\hfill & 2x-3\hfill \\ \hfill y& =\hfill & 2x-3\hfill \\ \hfill y& =\hfill & mx+b\hfill \\ \hfill m& =\hfill & 2\hfill \end{array}\hfill & & & & \begin{array}{}\\ \hfill y& =\hfill & 2x-3\hfill \\ \hfill y& =\hfill & mx+b\hfill \\ \hfill m& =\hfill & 2\hfill \end{array}\hfill \\ & & & \phantom{\rule{3.5em}{0ex}}y\text{-intercept is (0 ,−3)}\hfill & & & & y\text{-intercept is (0, −3)}\hfill \end{array}\)

The lines have the same slope, but they also have the same y-intercepts. Their equations represent the same line. They are not parallel; they are the same line.

This lesson is part of:

Graphs and Equations

View Full Tutorial

Track Your Learning Progress

Sign in to unlock unlimited practice exams, tutorial practice quizzes, personalized weak area practice, AI study assistance with Lexi, and detailed performance analytics.