Slope-Intercept Form

Slope-intercept form writes a linear relationship as y = mx + b, exposing its constant rate of change and starting value.

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Slope compares vertical change with horizontal change. A line contains two marked points joined by a right-angled rise-and-run triangle. The vertical and horizontal legs make the signed ratio behind slope visible.

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

Plot the intercept $(0,b)$.
Write slope as rise/run.
Move from the intercept using that rise and run.
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 $y=\tfrac45x+2$.

Rearranging other forms

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

$$ 2y=-4x+10, $$

$$ y=-2x+5. $$

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

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?