Slope-Intercept Form

A nonvertical line can be written $y=mx+b$, where $m$ is slope and $(0,b)$ is the $y$-intercept.

What the parameters reveal

The coefficient $m$ describes how much $y$ changes when $x$ increases by $1$. The constant $b$ is the value of $y$ when $x=0$.

For $y=3x-4$, the slope is $3$ and the $y$-intercept is $(0,-4)$. The line begins at height $-4$ on the vertical axis and rises $3$ for every step right.

Graphing from the equation

1. Plot the intercept $(0,b)$.
2. Write slope as rise/run.
3. Move from the intercept using that rise and run.
4. Plot another point and draw the line through the points.

You may also move left $3$ and up $2$; both directions stay on the same line.

Writing an equation from a graph

Read the vertical-axis crossing to find $b$. Then use two reliable points to calculate $m$. Substitute them into $y=mx+b$.

If a graph crosses at $(0,2)$ and rises $4$ while running $5$, then $m=4/5$ and the equation is $y=\tfrac45x+2$.

From a table

For equally spaced $x$-values, constant first differences in $y$ indicate a linear relationship. Divide change in $y$ by change in $x$ to find slope.

Here $m=(13-7)/(2-0)=3$, and $b=7$, so $y=3x+7$.

From two points

First calculate slope. Then substitute either point into $y=mx+b$ to find $b$.

Through $(2,5)$ and $(6,13)$, slope is $(13-5)/(6-2)=2$. Using $(2,5)$ gives $5=2(2)+b$, so $b=1$. The equation is $y=2x+1$.

Rearranging other forms

Isolate $y$. For $4x+2y=10$:

Dividing every term by the coefficient of $y$ is essential.

Modelling a context

Slope is the variable rate and intercept is the initial value. A phone plan costing $25$ dollars plus $8$ dollars per gigabyte can be modelled by

The variables may not be named $x$ and $y$; the structure is still slope-intercept form. Its domain should match the context.

Horizontal and vertical lines

A horizontal line has equation $y=b$ and slope $0$, so it fits $y=0x+b$. A vertical line $x=a$ cannot be written as $y=mx+b$ because its slope is undefined and one $x$-value corresponds to many $y$-values.

Common mistakes

Using the $x$-intercept as $b$. In $y=mx+b$, $b$ is the vertical-axis intercept.

Reading $-x$ as slope $0$. Its coefficient is $-1$.

Dividing only some terms when isolating $y$. Divide the entire equation.

Graphing slope from the origin automatically. Begin at $(0,b)$ unless $b=0$.

Quick self-check

• Is $y$ isolated?
• What are the units and meanings of $m$ and $b$?
• Does a second point satisfy my equation?
• Does the line’s direction match the sign of its slope?

Related topics

• Slope
• Linear Equation
• Point-Slope Form
• Introduction to Functions