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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 [link] .

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 = −2 x + 3 and 2 x + y = −1 on the same grid. The first equation is already in slope–intercept form: y = −2 x + 3 . We solve the second equation for y :

2 x + y = −1 y = −2 x 1

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?

y = m x + b y = m x + b y = −2 x + 3 y = −2 x 1 m = −2 m = −2 b = 3 , (0, 3) b = −1 , (0, −1)

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.

Use slopes and y -intercepts to determine if the lines 3 x 2 y = 6 and y = 3 2 x + 1 are parallel.

Solution

Solve the first equation for y . The equation is now in slope–intercept form. 3 x 2 y = 6 2 y = −3 x + 6 −2 y −2 = −3 x + 6 −2 y = 3 2 x 3 and y = 3 2 x + 1 The equation of the second line is already in slope–intercept form. Identify the slope and y -intercept of both lines. y = 3 2 x 3 y = m x + b m = 3 2 y = 3 2 x + 1 y = 3 2 x + 1 y = m x + b m = 3 2 y -intercept is (0, −3) y -intercept is (0, 1)

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.

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Use slopes and y -intercepts to determine if the lines 2 x + 5 y = 5 and y = 2 5 x 4 are parallel.

parallel

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Use slopes and y -intercepts to determine if the lines 4 x 3 y = 6 and y = 4 3 x 1 are parallel.

parallel

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

Solution

Write each equation in slope–intercept form. Since there is no x term we write 0 x . Identify the slope and y -intercept of both lines. y = −4 and y = 3 y = 0 x 4 y = 0 x + 3 y = 0 x 4 y = 0 x + 3 y = m x + b y = m x + b m = 0 m = 0 y -intercept is (0, 4) y -intercept is (0, 3)

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 ( 0 , −4 ) and ( 0 , 3 ) . The lines have the same slope and different y -intercepts and so they are parallel.

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Use slopes and y -intercepts to determine if the lines y = 8 and y = −6 are parallel.

parallel

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Use slopes and y -intercepts to determine if the lines y = 1 and y = −5 are parallel.

parallel

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

Solution

x = −2 and 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.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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