How do you find the x and y intercepts of a linear equation?

How do you find the x and y intercepts of a linear equation?

Intercepts are the points where a linear graph crosses the coordinate axes. Finding them is a fundamental skill in algebra for graphing lines quickly and understanding their behavior.

Definitions

• x-intercept: The point where the line crosses the x-axis. At this point, the y-coordinate is always 0, written as the coordinate pair $(x, 0)$.
• y-intercept: The point where the line crosses the y-axis. At this point, the x-coordinate is always 0, written as the coordinate pair $(0, y)$.

Step-by-Step Guide

1. Finding the y-intercept

To find where the line crosses the vertical y-axis, substitute $0$ for $x$ in the equation and solve for $y$.

• Shortcut: If the linear equation is in slope-intercept form ($y = mx + b$), the constant $b$ is automatically the y-intercept.

2. Finding the x-intercept

To find where the line crosses the horizontal x-axis, substitute $0$ for $y$ in the equation and solve for $x$.

Example

Find the intercepts of the equation: $2x + 3y = 12$.

• Find the y-intercept (set $x = 0$):

$$2(0) + 3y = 12$$

$$3y = 12$$

$$y = 4$$

Y-intercept: $(0, 4)$

• Find the x-intercept (set $y = 0$):

$$2x + 3(0) = 12$$

$$2x = 12$$

$$x = 6$$

X-intercept: $(6, 0)$