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The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.

Now, we’ll consider a vertical line, the line.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).

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 not defined. So we say that the slope of the vertical line x = 3 is undefined.

The slope of any vertical line is undefined. When the x -coordinates of a line are all the same, 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:

x = 8 y = −5 .

Solution

x = 8
This is a vertical line.
Its slope is undefined.

y = −5
This is a horizontal line.
It has slope 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

This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.

Remember, we ‘read’ a line from left to right, just like we read written words in English.

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

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.

Sometimes we’ll need to find the slope of a line between two points when we don’t 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 as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some 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 the points.

( x 1 , y 1 ) read ‘ x sub 1, y sub 1’ ( x 2 , y 2 ) read ‘ x sub 2, y sub 2’

The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?

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.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points ( 2 , 3 ) and ( 7 , 6 ) .

The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.

Since we have two points, we will use subscript notation, ( 2 , x 1 , 3 y 1 ) ( 7 , 6 x 2 , y 2 ) .

On the graph, we counted the rise of 3 and the run of 5.

Notice that the rise of 3 can be found by subtracting the y -coordinates 6 and 3.

3 = 6 3

And the run of 5 can be found by subtracting the x -coordinates 7 and 2.

5 = 7 2

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 y 2 , the y -coordinate of the second point and 3 is y 1 , the y -coordinate of the first point.

So we can rewrite the slope using subscript notation. m = y 2 y 1 7 2

Also, 7 is x 2 , the x -coordinate of the second point and 2 is x 1 , the x -coordinate of the first point.

So, again, we rewrite the slope 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.

Questions & Answers

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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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