4.4 Understand slope of a line  (Page 5/14)

Solution

$\begin{array}{cccc}\text{We’ll call}\left(1,2\right)\text{point \#1 and}\left(4,5\right)\text{point \#2.}\hfill & & & \left(\stackrel{x_1,y_1}{{\displaystyle 1,2}}\right){\left(\stackrel{x_2,y_2}{4,5}\right)}_{}\hfill \\ \text{Use the slope formula.}\hfill & & & m=\frac{y_2-y_1}{x_2-x_1}\hfill \\ \text{Substitute the values.}\hfill & & \\ y\text{of the second point minus}y\text{of the first point}\hfill & & & m=\frac{5-2}{x_2-x_1}\hfill \\ x\text{of the second point minus}x\text{of the first point}\hfill & & & m=\frac{5-2}{4-1}\hfill \\ \text{Simplify the numerator and the denominator.}\hfill & & & m=\frac{3}{3}\hfill \\ \text{Simplify.}\hfill & & & m=1\hfill \end{array}$

Let’s confirm this by counting out the slope on a graph using $m=\frac{\text{rise}}{\text{run}}$ .

It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.

Solution

$\begin{array}{ccccc}\text{We’ll call}\left(\mathrm{-2},\mathrm{-3}\right)\text{point \#1 and}\left(\mathrm{-7},4\right)\text{point \#2.}\hfill & & & & \left(\stackrel{x_1,y_1}{{\displaystyle \mathrm{-2},\mathrm{-3}}}\right)\left(\stackrel{x_2,y_2}{{\displaystyle \mathrm{-7},4}}\right)\hfill \\ \text{Use the slope formula.}\hfill & & & & m=\frac{y_2-y_1}{x_2-x_1}\hfill \\ \text{Substitute the values.}\hfill & & & & \\ y\text{of the second point minus}y\text{of the first point}\hfill & & & & m=\frac{4-\left(\mathrm{-3}\right)}{x_2-x_1}\hfill \\ x\text{of the second point minus}x\text{of the first point}\hfill & & & & m=\frac{4-(\mathrm{-3})}{\mathrm{-7}-(\mathrm{-2})}\hfill \\ \text{Simplify.}\hfill & & & & m=\frac{7}{\mathrm{-5}}\hfill \\ & & & & m=-\frac{7}{5}\hfill \end{array}$

Let’s verify this slope on the graph shown.

Solution

Solution

Plot the given point, the y -intercept, $(0,2)$ .

$\begin{array}{ccccc}\text{Identify the rise and the run.}\hfill & & \hfill m& =\hfill & -\frac{2}{3}\hfill \\ & & \hfill \frac{\text{rise}}{\text{run}}& =\hfill & \frac{\mathrm{-2}}{3}\hfill \\ & & \hfill \text{rise}& =\hfill & \mathrm{-2}\hfill \\ & & \hfill \text{run}& =\hfill & 3\hfill \end{array}$

Count the rise and the run. Mark the second point.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is $m=-\frac{2}{3}$ , it can be written as $m=\frac{2}{\mathrm{-3}}$ . Go back to $\left(0,2\right)$ and count out the rise, 2, and the run, $\mathrm{-3}$ .