Lesson 1: 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

We can write the equation of any non-vertical line in slope-intercept form by solving the equation for $y$ in terms of $x$.

Caution: Do not confuse solving for $y$ with finding the $y$-intercept. When we solve for $y$, we are writing the equation in another form, so both variables, $x$ and $y$, still appear in the equation.

Examples

• To convert $6x + 3y = 12$, subtract $6x$ from both sides to get $3y = -6x + 12$. Then divide all terms by $3$ to get the final form $y = -2x + 4$.
• For the equation $5x - 2y = 10$, subtract $5x$ to get $-2y = -5x + 10$. Divide everything by $-2$ to find the slope-intercept form, $y = \frac{5}{2}x - 5$.
• To solve $x + 4y = 8$ for $y$, subtract $x$ from both sides giving $4y = -x + 8$. Then divide by $4$ to get $y = -\frac{1}{4}x + 2$.