Slope-Intercept Form — Mathematics Knowledge System

Overview

Slope-intercept form writes the equation of a line as y = mx + b, where m is the slope and b is the y-intercept. It is the most convenient form for graphing a line quickly, since both the starting point (the y-intercept) and the direction (the slope) can be read off directly from the equation without any computation.

Intuition

Think of y = mx + b as a recipe: b tells you where to start on the y-axis, and m tells you how to move from there — for every 1 step right, move m steps up (or down, if m is negative). Because both pieces of information are visible directly in the equation, slope-intercept form turns graphing into a two-step mechanical process rather than a calculation.

Interactive Graph

Formal Definition

Definition

The slope-intercept form of a line is:

y = mx+b

m is the slope; b is the y-intercept (the point (0,b) where the line crosses the y-axis)

Slope-intercept form

Notation

Notation	Meaning
$m$	Slope of the line
$b$	y-intercept — the value of y when x = 0

Properties

y-intercept

\text{Setting } x=0 \text{ gives } y=b, \text{ so the line crosses the y-axis at } (0,b)

Conversion from standard form

Ax+By=C \;\Rightarrow\; y=-\frac{A}{B}x+\frac{C}{B} \quad (B\neq 0)

Parallel lines share slope

y=mx+b_1 \parallel y=mx+b_2 \text{ whenever } b_1 \neq b_2

Applications

A savings account with a flat monthly deposit models balance as B(t) = mt + b, where b is the initial deposit and m is the monthly contribution.

Worked Examples

Substitute m=3, b=-2 directly into y=mx+b.
$y=3x-2$

Answer: y = 3x - 2

Practice Problems

Difficulty 2/10

Find the slope and y-intercept of y = -5x + 8.

Difficulty 3/10

Convert 3x - 6y = 12 into slope-intercept form.

Common Mistakes

Common Mistake

Confusing which coefficient in y = mx + b is the slope and which is the intercept.

m (the coefficient of x) is ALWAYS the slope; b (the constant term) is ALWAYS the y-intercept. Double check by plugging in x=0: y should equal b.

Common Mistake

Forgetting to divide EVERY term by the coefficient of y when converting from standard form, e.g. converting 2y = 4x + 6 to y = 2x + 6 instead of y = 2x + 3.

When isolating y, divide ALL terms on both sides by y's coefficient: 2y=4x+6 → y = (4x+6)/2 = 2x+3, not 2x+6.

Summary

Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
The y-intercept (0, b) and slope m can be read directly from the equation.
To convert from standard form Ax + By = C, solve for y.
Lines with the same m but different b are parallel and never intersect.

References

WebsiteWikipedia — Linear equation (Slope-intercept form)