1.10.2 Graphing Equations in Context

Find the Slope of a Line

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter.

In mathematics, the measure of the steepness of a line is called the of slope the line. The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. As you ski or jog down a hill, you definitely experience a slope. We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope is a rate of change.

Slope of a Line

The slope of a line is $m=\frac{rise}{run}$. The rise measures the vertical change and the run measures the horizontal change. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical. To find the slope of the line, we measure the vertical distance is called the rise and the horizontal distance is called the run.

How To Find the Slope from a Graph

Find the slope of a line from its graph using $m=\frac{rise}{run}$

Step 1 - Locate two points on the line whose coordinates are integers.

Step 2 - Starting with one point, sketch a right triangle, going from the first point to the second point.

Step 3 - Count the rise and the run on the legs of the triangle.

Step 4 - Take the ratio of rise to run to find the slope: $m=\frac{rise}{run}$.

Example 1

Find the slope of the line shown.

Solution

Step 1 - Locate two points on the graph whose coordinates are integers.

$(0,5)$ and $(3,3)$

Step 2 - Starting at $(0,5)$, sketch a right triangle to $(3,3)$ as shown in this graph.

Step 3 - Count the rise(how many spaces up or down)— since it goes down, it is negative.

The rise is $-2$.

Step 4 - Count the run (how many spaces to the right or left).

The run is 3.

Step 5 - Use the slope formula.

$m=\frac{rise}{run}$

Step 6 - Substitute the values of the rise and run.

$m=\frac{-2}{3}$

Step 7 - Simplify.

$m=-\frac{2}{3}$

So $y$ decreases by 2 units as $x$ increases by 3 units.

Slopes of Horizontal and Vertical Lines

How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line, $y=4$, we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.

Step 1 - What is the rise?

The rise is 0.

Step 2 - What is the run?

The run is 3.

Step 3 - What is the slope?

$m=\frac{rise}{run}$

$m=\frac{0}{3}$

$m=0$

So The slope of the horizontal line $y=4$ is 0.

Let's also consider a vertical line, the line $x=3$, as shown in the graph below.

Step 1 - What is the rise?

The rise is 2.

Step 2 - What is the run?

The run is 0.

Step 3 - What is the slope?

$m=\frac{rise}{run}$

$m=\frac{2}{0}$

The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line is undefined.

• All horizontal lines have slope 0. When the $y$–coordinates are the same, the rise is 0.
• The slope of any vertical line is undefined. When the $x$–coordinates of a line are all the same, the run is 0.

Quick Guide to the Slopes of Lines

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.

We will use ($x$₁ ,$y$₁) to identify the first point and ($x$₂,$y$₂) to identify the second point. 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), as shown in this graph.

Step 1 - Since we have two points, we will use subscript notation.

$\left(\genfrac{}{}{0ex}{}{x_1,y_1}{2,3}\right)$$\left(\genfrac{}{}{0ex}{}{x_2,y_2}{7,6}\right)$

Step 2 - What is the slope?

The rise is 3.

The run is 5.

Step 3 - What is the slope?

$m=\frac{rise}{run}$

On the graph, we counted the rise of 3 and the run of 5. $m=\frac{3}{5}$

Notice that the rise of 3 can be found by subtracting the $y$–coordinates, 6 and 3, and the run of 5 can be found by subtracting the $x$–coordinates, 7 and 2.

Step 4 - We rewrite the rise and run by putting in the coordinates.

$m=\frac{6-3}{7-2}$

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.

$m=\frac{y_2-y_1}{7-2}$

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

$m=\frac{y_2-y_1}{x_2-x_1}$

We’ve shown that $m=\frac{y_2-y_1}{x_2-x_1}$ is really another version of $m=\frac{rise}{run}$. We can use this formula to find the slope of a line when we have two points on the line.

How to Find the Slope from Two Points

The slope of the line between two points ($x_1,y_1)$ and ($x_2,y_2$)  is: $m=\frac{y_2-y_1}{x_2-x_1}$

Example 2

Step 1 - We’ll call (-2, -3) point #1 and (-7, 4) point #2.

$\left(\genfrac{}{}{0ex}{}{x_1,y_1}{-2,-3}\right)$ $\left(\genfrac{}{}{0ex}{}{x_2,y_2}{-7,4}\right)$

Step 2 - Use the slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}$

Step 3 - Substitute the values.

$y$ of the second point minus $y$ of the first point.

$x$ of the second point minus $x$ of the first point

$m=\frac{4-(-3)}{-7-(-2)}$

Step 4 - Simplify.

$m=\frac{4+3}{-7+2}$

$m=\frac{7}{-5}$

$m=-\frac{7}{5}$

Step 5 - Let's verify this slope on the graph shown.

$m=\frac{y_2-y_1}{x_2-x_1}$

Try it

Try It: Find the Slope of a Line

Find the slope of the line graph below.

Compare your answer: Your answer may vary, but here is a sample.

Step 1 - Locate two points on the graph whose coordinates are integers.

$(0,-5)(4,1)$

Step 2 - Starting at (0,-5), sketch a right triangle to (4,1) as shown in this graph.

Step 3 - Count the rise (how many spaces up or down)–since it goes up, it is positive.

The rise is -6.

Step 4 -Count the run (how many spaces to the right or left).

The run is 4.

Step 5 - Use the slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{1-(-5)}{4-0}$

$m=\frac{1+5}{4-0}$

$m=\frac{6}{4}$

Step 4 - Simplify.

$m=\frac{3}{2}$

So $y$ increases by 3 units as $x$ increases by 2 units.