To graph linear equations using intercepts, you first need to know about the x-intercept and y-intercept.

A common way to write a linear equation is in slope-intercept form:

$$y = mx + b$$

In this equation, m is the slope, and b is the y-intercept. The y-intercept is where the line crosses the y-axis. This happens when x = 0.

On the other hand, the x-intercept is where the line crosses the x-axis, and that occurs when y = 0.

How to Find the Intercepts

1. Finding the Y-Intercept:

   • To find the y-intercept, set x = 0 in the equation.
   • For example, if the equation is y = 2x + 3:
     • Substitute x = 0:
     $$y = 2(0) + 3 = 3$$
   • So, the y-intercept is (0, 3).

2. Finding the X-Intercept:

   • To find the x-intercept, set y = 0 in the equation.
   • Using the same equation, y = 2x + 3:
     • Substitute y = 0:
     $$0 = 2x + 3$$
     • Now solve for x:
     $$2x = -3 \implies x = -\frac{3}{2}$$
   • Therefore, the x-intercept is (-1.5, 0).

Plotting the Intercepts

After you find the intercepts, you can place these points on the graph:

• Plot the y-intercept (0, 3) on the y-axis.
• Plot the x-intercept (-1.5, 0) on the x-axis.

Drawing the Line

Once you have both points plotted, draw a straight line connecting them. Make sure the line goes in both directions and label it with the equation of the line.

Example Equation

Let's look at the equation 3x - 4y = 12. Here’s how to find the intercepts:

• Y-Intercept:

  • Set x = 0:
  $$-4y = 12 \implies y = -3 \quad \text{(Point: (0, -3))}$$

• X-Intercept:

  • Set y = 0:
  $$3x = 12 \implies x = 4 \quad \text{(Point: (4, 0))}$$

Summary

Knowing the coordinates of the intercepts helps you easily create a graph of any linear equation. By finding both intercepts and drawing a line through them, you can show the solutions of the linear equation on a graph.