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Understanding the Slope of a Line

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

\(\text { Horizontal line } y=b\) \( \text { Vertical line } x=a \)
\( y \text { -coordinates are the same.}\) \( x \text { -coordinates are the same.}\)


So how do we find the slope of the horizontal line \(y =4\)? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let's see what happens when we do this.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A li

What is the rise? The rise is \(0\).
Count the run. The run is \(3\).
What is the slope? \begin{align} \begin{array}{l} m=\frac{\text { rise }}{\text { run }} \\ m=\frac{0}{3} \\ m=0 \end{array} \end{align}
  The slope of the horizontal line \(y =4\)  is \(0\).

Table 4.34

All horizontal lines have slope \(0\). When the \(y\)-coordinates are the same, the rise is \(0\).

Slope of a Horizontal Line

The slope of a horizontal line, \(y =b\), is \(0\).

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 li

What is the rise? The rise is \(2\).
Count the run. The run is \(0\).
What is the slope? \(\begin{align} \begin{array}{l} m=\frac{\text { rise }}{\text { run }} \\ m=\frac{2}{0} \end{array} \end{align}\)

Table 4.35

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.

Example 4.32

Find the slope of each line:

  1. \(x =8\)

  2. \(y =−5\)

Solution
  1. \(x =8\)
    This is a vertical line.
    Its slope is undefined.

  2. \(y =−5\)
    This is a horizontal line.
    It has slope \(0\).
Try It 4.63

Find the slope of the line: \(x =−4\).

Try It 4.64

Find the slope of the line: \(y =7\).

 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

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