Intercepts
Simple Explanation
An intercept is where a line crosses an axis. The y-intercept is where it crosses the y-axis (where x = 0); the x-intercept is where it crosses the x-axis (where y = 0).
Why Do We Need It?
Intercepts are often the easiest, most meaningful points on a line to find and interpret — like a starting value or a break-even point.
See It
A coordinate plane showing a line crossing the y-axis at (0, 6) and the x-axis at (4, 0), both marked
Formula
Equation of a Straight Line
y = mx + c
Describes every point on a non-vertical straight line, using its slope and y-intercept.
- m
- — the slope of the line
- c
- — the y-intercept — where the line crosses the y-axis
- x, y
- — the coordinates of any point on the line
When to use it: Whenever you need to write down, graph, or predict values along an entire straight line, not just one point on it.
Worked Example
Find both intercepts of a line
Find the x-intercept and y-intercept of the line y = 2x − 8.
Why Does This Work?
Every point on the y-axis has x = 0, and every point on the x-axis has y = 0 — substituting those fixed values into the line's equation finds exactly where it meets each axis.
Real-Life Example
Break-even point in a budget
A savings graph plots money saved (y) against months (x), starting from an initial debt.
The x-intercept shows the exact month the balance reaches zero — a real break-even point, found the same way as a mathematical x-intercept.
Practice
What is the y-intercept of the line y = 5x + 9?
EasyWhat is the x-intercept of the line y = 3x − 12? (Give just the x-value.)
MediumCommon mistake
Swapping which coordinate is set to zero — for the x-intercept it is y that's set to 0 (not x), and vice versa for the y-intercept.
Quick Review
- y-intercept: set x = 0, solve for y.
- x-intercept: set y = 0, solve for x.
- A non-horizontal, non-vertical line has exactly one x-intercept and one y-intercept.