Lesson 5: Graphing Linear Equations in Slope-Intercept Form

Property

To find the slope of a line from its graph:

1. Locate two points on the graph whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8 - 2 = 6$ and the run is $4 - 1 = 3$. The slope is $m = \frac{6}{3} = 2$.
• A line passes through $(0, 6)$ and $(2, 2)$. The rise is $2 - 6 = -4$ and the run is $2 - 0 = 2$. The slope is $m = \frac{-4}{2} = -2$.
• A line on a graph connects points $(-3, 1)$ and $(5, 7)$. The rise is $7-1=6$ and the run is $5-(-3)=8$. The slope is $m = \frac{6}{8} = \frac{3}{4}$.

Explanation

To find slope from a graph, pick two easy-to-read points. Count the vertical distance (rise) and horizontal distance (run) to get from one point to the other. The slope is simply the rise divided by the run.

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. This is the slope formula. The slope is the difference in the y-coordinates divided by the difference in the x-coordinates.

Examples

• For the points $(2, 5)$ and $(4, 11)$, the slope is $m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$.
• For the points $(-3, 6)$ and $(1, 4)$, the slope is $m = \frac{4 - 6}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}$.
• For the points $(5, -1)$ and $(-2, 3)$, the slope is $m = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}$.

Explanation

The slope formula is a tool to find a line's steepness without a graph. It calculates the rise by subtracting y-values ($y_2 - y_1$) and the run by subtracting x-values ($x_2 - x_1$), then divides them.