Understanding X-Intercepts and Y-Intercepts

When graphing a linear equation, two of the most important points are where the line crosses the axes on a coordinate plane. These points are called the intercepts. Because a straight line can be drawn using just two points, finding the intercepts is one of the easiest ways to graph a line.

What is an X-Intercept?

The $x$-intercept is the point where the line crosses the horizontal $x$-axis. Because the $x$-axis sits exactly at a height of zero, the $y$-value is always $0$ at the $x$-intercept. It is written as the coordinate point $(x, 0)$.

What is a Y-Intercept?

The $y$-intercept is the point where the line crosses the vertical $y$-axis. Because the $y$-axis is located exactly at the center horizontally, the $x$-value is always $0$ at the $y$-intercept. It is written as the coordinate point $(0, y)$.

How to Find Intercepts

To find the intercepts from an algebraic equation, you simply plug in $0$ for one variable and solve for the other.

Example: Find the $x$-intercept and $y$-intercept of $2x + 3y = 6$.

1. Find the $x$-intercept (Set $y = 0$): $2x + 3(0) = 6$ $2x = 6$ $x = 3$ The $x$-intercept is $(3, 0)$.

2. Find the $y$-intercept (Set $x = 0$): $2(0) + 3y = 6$ $3y = 6$ $y = 2$ The $y$-intercept is $(0, 2)$.

Graphing a Line Using Intercepts

Once you have both intercepts, graphing the line is incredibly simple.

Example: Graph $x - 2y = 4$ using its intercepts.

Step 1: Find the intercepts.

• $x$-intercept: Set $y = 0$. $x - 2(0) = 4 \implies x = 4$ The point is $(4, 0)$.
• $y$-intercept: Set $x = 0$. $0 - 2y = 4 \implies -2y = 4 \implies y = -2$ The point is $(0, -2)$.

Step 2: Plot and draw. Plot the point $(4, 0)$ on the horizontal $x$-axis. Then, plot the point $(0, -2)$ on the vertical $y$-axis. Finally, use a straightedge to draw a straight line directly through both points. You have successfully graphed the line!