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The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line passes through the labeled points “ordered pair 3, 2” and “ordered pair 3, 0”.
What is the rise? The rise is 2.
What is the run? The run is 0.
What is the slope? m = rise run
m = 2 0

But we can’t divide by 0 . Division by 0 is undefined. So we say that the slope of the vertical line x = 3 is undefined. The slope of all vertical lines is undefined, because the run is 0 .

Slope of a vertical line

The slope of a vertical line    , x = a , is undefined.

Find the slope of each line:

  1. x = 8
  2. y = −5

Solution

x = 8

This is a vertical line, so its slope is undefined.

y = −5

This is a horizontal line, so its slope is 0 .

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Find the slope of the line: x = −4 .

undefined

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Find the slope of the line: y = 7 .

0

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Quick guide to the slopes of lines

The figure shows 4 arrows. The first rises from left to right with the arrow point upwards. It is labeled “positive”. The second goes down from left to right with the arrow pointing downwards. It is labeled “negative”. The third is horizontal with arrow heads on both ends. It is labeled “zero”. The last is vertical with arrow heads on both ends. It is labeled “undefined.”

Use the slope formula to find the slope of a line between two points

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.

Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair     ( x , y ) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol ( x , y ) be used to represent two different points?

Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.

  • ( x 1 , y 1 ) read x sub 1 , y sub 1
  • ( x 2 , y 2 ) read x sub 2 , y sub 2

We will use ( x 1 , y 1 ) to identify the first point and ( x 2 , y 2 ) to identify the second point. If we had more than two points, we could use ( x 3 , y 3 ) , ( x 4 , y 4 ) , and so on.

To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points ( 2 , 3 ) and ( 7 , 6 ) in [link] .

The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 7. A line runs through the labeled points 2, 3 and 7, 6. A line segment runs from the point 2, 3 to the unlabeled point 2, 6. It is labeled y sub 2 minus y sub 1, 6 minus 3, 3. A line segment runs from the point 7, 6 to the unlabeled point 2, 6.  It os labeled x sub 2 minus x sub 1, 7 minus 2, 5.

Since we have two points, we will use subscript notation.

( 2 , 3 ) x 1 , y 1 ( 7 , 6 ) x 2 , y 2

On the graph, we counted the rise of 3 . The rise can also be found by subtracting the y -coordinates of the points.

y 2 y 1 6 3 3

We counted a run of 5 . The run can also be found by subtracting the x -coordinates .

x 2 x 1 7 2 5
We know m = rise run
So m = 3 5
We rewrite the rise and run by putting in the coordinates. m = 6 3 7 2
But 6 is the y -coordinate of the second point, y 2
and 3 is the y -coordinate of the first point y 1 .
So we can rewrite the rise using subscript notation.
m = y 2 y 1 7 2
Also 7 is the x -coordinate of the second point, x 2
and 2 is the x -coordinate of the first point x 2 .
So we rewrite the run using subscript notation.
m = y 2 y 1 x 2 x 1

We’ve shown that m = y 2 y 1 x 2 x 1 is really another version of m = rise run . We can use this formula to find the slope of a line when we have two points on the line.

Slope formula

The slope of the line between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is

m = y 2 y 1 x 2 x 1

Say the formula to yourself to help you remember it:

Slope is y of the second point minus y of the first point
over
x of the second point minus x of the first point.

Doing the Manipulative Mathematics activity “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points.

Find the slope of the line between the points ( 1 , 2 ) and ( 4 , 5 ) .

Solution

We’ll call ( 1 , 2 ) point #1 and ( 4 , 5 ) point #2. ( 1 , 2 ) x 1 , y 1 and ( 4 , 5 ) x 2 , y 2
Use the slope formula. m = y 2 y 1 x 2 x 1
Substitute the values in the slope formula:
y of the second point minus y of the first point m = 5 2 x 2 x 1
x of the second point minus x of the first point m = 5 2 4 1
Simplify the numerator and the denominator. m = 3 3
m = 1

Let’s confirm this by counting out the slope on the graph.
The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -1 to 7. Two labeled points are drawn at  “ordered pair 1, 2” and  “ordered pair 4, 5”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, 2” and “ordered pair 1, 5 ”.  It is labeled “rise”. A horizontal line segment connects “ordered pair 1, 5” and “ordered pair 4, 5”. It is labeled “run”.

The rise is 3 and the run is 3 , so

m = rise run m = 3 3 m = 1

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Questions & Answers

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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