4.4 Understand slope of a line  (Page 3/14)

Solution

Locate two points on the graph whose coordinates are integers.	$\left(0,5\right)$ and $\left(3,3\right)$
Which point is on the left?	$\left(0,5\right)$
Starting at $\left(0,5\right)$ , sketch a right triangle to $\left(3,3\right)$ .
Count the rise—it is negative.	The rise is $\mathrm{-2}$ .
Count the run.	The run is 3.
Use the slope formula.	$m=\frac{\text{rise}}{\text{run}}$
Substitute the values of the rise and run.	$m=\frac{\mathrm{-2}}{3}$
Simplify.	$m=-\frac{2}{3}$
	The slope of the line is $-\frac{2}{3}$ .

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

What if we used the points $\left(\mathrm{-3},7\right)$ and $\left(6,1\right)$ to find the slope of the line?

The rise would be $\mathrm{-6}$ and the run would be 9. Then $m=\frac{\mathrm{-6}}{9}$ , and that simplifies to $m=-\frac{2}{3}$ . Remember, it does not matter which points you use—the slope of the line is always the same.

Solution

Locate two points on the graph whose coordinates are integers.	$\left(2,3\right)$ and $\left(7,6\right)$
Which point is on the left?	$\left(2,3\right)$
Starting at $\left(2,3\right)$ , sketch a right triangle to $\left(7,6\right)$ .
Count the rise.	The rise is 3.
Count the run.	The run is 5.
Use the slope formula.	$m=\frac{\text{rise}}{\text{run}}$
Substitute the values of the rise and run.	$m=\frac{3}{5}$
	The slope of the line is $\frac{3}{5}$ .

This means that $y$ increases 5 units as $x$ increases 3 units.

When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.

What would happen if we started with the point on the right?

Let’s use the points $\left(2,3\right)$ and $\left(7,6\right)$ again, but now we’ll start at $\left(7,6\right)$ .

$\begin{array}{ccc}\text{Count the rise.}\hfill & & \text{The rise is}\mathrm{-3}.\hfill \\ \begin{array}{c}\text{Count the run. It goes from right to left, so}\hfill \\ \text{it is negative.}\hfill \end{array}\hfill & & \text{The run is}\mathrm{-5}.\hfill \\ \text{Use the slope formula.}\hfill & & m=\frac{\text{rise}}{\text{run}}\hfill \\ \text{Substitute the values of the rise and run.}\hfill & & m=\frac{\mathrm{-3}}{\mathrm{-5}}\hfill \\ & & \text{The slope of the line is}\frac{3}{5}.\hfill \end{array}$

It does not matter where you start—the slope of the line is always the same.