Section 4.6: Writing Equations in Slope-Intercept Form

Property

A linear equation written in the form

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$-intercept.

Examples

• The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$-intercept is $(0, 5)$.
• For $y = -2x - 1$, the slope is $-2$ and the $y$-intercept is $(0, -1)$.
• In the equation $y = \frac{1}{2}x + 4$, the slope is $\frac{1}{2}$ and the $y$-intercept is $(0, 4)$.

Explanation

This form is a recipe for drawing a line. The '$b$' tells you your starting point on the y-axis, and the '$m$' (slope) gives you directions on how steep to draw the line from there.

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.