Slope-Intercept Form
What is the Slope-Intercept Form?
Slope-intercept form immediately tells you two key facts about a line just by reading its equation: how steep it is (m) and where it crosses the y-axis (b). That makes it the most convenient of the several standard ways to write a line's equation for graphing quickly — plot the y-intercept first, then use the slope to find a second point by counting rise over run.
If you're instead given two points on the line rather than m and b directly, find the slope first with m = (y₂ − y₁) / (x₂ − x₁), then substitute one point into y = mx + b to solve for b. This two-step process (slope, then intercept) is the most common real use of the formula in practice, since problems rarely hand you m and b directly.
The sign and size of m carry real meaning beyond just "steepness": a slope of 2 means y increases twice as fast as x, a slope of −1 means y decreases exactly as fast as x increases, and a slope of 0 means a perfectly horizontal line — y never changes no matter what x is. Slope-intercept form is also the version of a line's equation used most often outside pure math class, since m and b map directly onto real quantities like a rate of change (m) and a starting value (b) in word problems about cost, distance, or growth over time.
What Each Variable Means
When to Use It
- Graphing a line quickly from its equation
- Writing the equation of a line when the slope and y-intercept are known or easy to find
- Comparing two lines' steepness or predicting where they'll cross the y-axis
Step-by-Step Example
Problem: A line has slope 2 and y-intercept −3. Write its equation and find y when x = 4.
Read the slope and y-intercept from the problem.
m = 2, b = -3Substitute m and b into y = mx + b.
y = 2x - 3Plug in the given x value.
y = 2(4) - 3 = 8 - 3Interactive Calculator
Common Mistakes
Mistake: Confusing the slope and y-intercept when reading an equation.
Fix: In y = mx + b, m is always the coefficient of x (the slope), and b is the standalone constant (the y-intercept) — the order never changes.
Mistake: Forgetting the sign of a negative y-intercept.
Fix: y = mx + b with b = -3 must be written as y = mx − 3, not y = mx + 3.
Practice Questions
Write the equation of a line with slope −1 and y-intercept 5, then find y when x = 2.
A line passes through (0, 4) and (2, 8). Find its slope-intercept equation.
Hint: The y-intercept is given directly by the point (0, 4); find the slope from the two points first.
Frequently Asked Questions
What if the line is vertical?
A vertical line (x = constant) has an undefined slope and cannot be written in slope-intercept form at all — it isn't a function of x.
How do I find the slope from two points?
m = (y₂ − y₁) / (x₂ − x₁). Once you have m, substitute either point into y = mx + b and solve for b.