Graphing Linear Equations

Graphing Linear Equations

There are several reliable methods for graphing a line from its equation. This lesson covers three of the most useful: a table of values, slope-intercept form, and intercepts.

Method 1: Table of Values

Pick a few simple values for x, substitute into the equation to find the matching y, then plot the resulting points and connect them with a straight line. For y = 2x - 1:

x: -1, 0, 1, 2    y: -3, -1, 1, 3

Plotting (-1,-3), (0,-1), (1,1), (2,3) and connecting them gives the graph of the line. This method always works, even for equations not in slope-intercept form.

Method 2: Slope-Intercept Form

If the equation is (or can be rewritten as) y = mx + b, plot the y-intercept (0, b) first. Then use the slope m — written as a fraction \tfrac{\text{rise}}{\text{run}} — to find a second point: move "run" units horizontally and "rise" units vertically from the intercept. Connect the two points with a straight line.

For y = \tfrac{2}{3}x + 1: start at (0, 1), then move 3 units right and 2 units up to reach (3, 3), and draw the line through both points.

Method 3: Using Intercepts

The x-intercept is where the line crosses the x-axis (where y = 0), and the y-intercept is where it crosses the y-axis (where x = 0). Since two points determine a line, this gives you a fast way to graph an equation like 3x + 2y = 12:

• Set y = 0: 3x = 12, so x = 4. The x-intercept is (4, 0).
• Set x = 0: 2y = 12, so y = 6. The y-intercept is (0, 6).

Plot (4,0) and (0,6), then draw the line connecting them.

Example

Graph 2x - y = 4 using the intercept method.

1. Set y=0: 2x = 4, so x = 2. x-intercept: (2, 0).
2. Set x=0: -y = 4, so y = -4. y-intercept: (0, -4).
3. Plotting and connecting (2,0) and (0,-4) gives a line that rises steeply from lower-left to upper-right — consistent with the equation's slope of m = \dfrac{0-(-4)}{2-0} = 2.

Key Takeaways

• A table of values always works, even when an equation isn't in slope-intercept form.
• From slope-intercept form, plot the y-intercept, then use rise/run to find a second point.
• Finding the x- and y-intercepts (setting the other variable to 0) is often the fastest method for equations like Ax + By = C.
• Only two points are needed to graph a line, but a third point is a great way to check your work.