Lesson 9: Analyze Linear Equations: y = mx + b

Property

The slope-intercept form for a linear equation is $y = mx + b$, where $m$ is the slope of the line and the point $(0, b)$ is the y-intercept.

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Property

To graph a line using the slope-intercept method:
a. Plot the y-intercept $(0, b)$.
b. Use the definition of slope, $m = \frac{\Delta y}{\Delta x}$, to find a second point on the line. Starting at the y-intercept, move $\Delta y$ units in the y-direction and $\Delta x$ units in the x-direction. Plot a second point at this location.
c. Use an equivalent form of the slope to find a third point, and draw a line through the points.

Examples

• To graph $y = 1 + 2x$, start by plotting the y-intercept at $(0, 1)$. The slope is $m = 2 = \frac{2}{1}$, so from $(0, 1)$, move up 2 units and right 1 unit to find the next point, $(1, 3)$.
• To graph $y = 3 - \frac{1}{2}x$, begin at the y-intercept $(0, 3)$. The slope is $m = -\frac{1}{2} = \frac{-1}{2}$, so from $(0, 3)$, move down 1 unit and right 2 units to plot the point $(2, 2)$.
• To graph $y = -4 + \frac{5}{3}x$, plot the y-intercept at $(0, -4)$. The slope is $m = \frac{5}{3}$, so move up 5 units and right 3 units from $(0, -4)$ to find the point $(3, 1)$.

Explanation

This is a two-step method for drawing lines. First, plot your starting point at the y-intercept $(0, b)$. Then, use the slope $m = \frac{\text{rise}}{\text{run}}$ as a map to find your next point, and connect them to draw the line.