Graphing Linear Equations — Mathematics Knowledge System

Overview

Graphing a linear equation means drawing the straight line that represents all of its solutions in the coordinate plane. Because a line is entirely determined by two points (or one point and a slope), graphing a linear equation reduces to finding just enough information — an intercept, a slope, or two solution pairs — to draw the line accurately.

Intuition

A line has no curves or bends, so once you know any two points on it, you know the entire line — just connect them and extend in both directions. The fastest way to find two easy points is often the x- and y-intercepts (set the other variable to 0), or, if the equation is in slope-intercept form, start at the y-intercept and use the slope as a step-by-step direction to plot a second point.

Interactive Graph

Formal Definition

Definition

Two standard graphing strategies for a linear equation:

y=mx+b: \text{ plot } (0,b), \text{ then move } \Delta y = m \text{ (rise) and } \Delta x=1 \text{ (run) to a second point}

Slope-intercept method

Ax+By=C: \text{ x-intercept } \left(\tfrac{C}{A},0\right), \text{ y-intercept } \left(0,\tfrac{C}{B}\right)

Intercept method

Properties

Two points determine a line

\text{Any two distinct points uniquely determine a straight line}

x-intercept

\text{Set } y=0 \text{ and solve for } x

y-intercept

\text{Set } x=0 \text{ and solve for } y

Applications

Graphing supply and demand as linear equations lets analysts visually locate the market equilibrium at their intersection.

Worked Examples

Plot the y-intercept (0, -3).
$(0,-3)$
From there, use the slope 2 = 2/1: move right 1, up 2, to reach the next point.
$(1,-1)$
Draw a straight line through both points, extending in both directions.

Answer: Line through (0, -3) and (1, -1)

Practice Problems

Difficulty 2/10

Find the x- and y-intercepts of 5x - 2y = 20 to graph it.

Difficulty 3/10

Using slope-intercept form, list two points you would plot to graph y = -3x + 4.

Common Mistakes

Common Mistake

Plotting the y-intercept as (b, 0) instead of (0, b).

The y-intercept is always where the line crosses the y-axis, meaning x = 0. The point is (0, b), not (b, 0).

Common Mistake

Using the intercept method on an equation where an intercept doesn't exist or is at the origin, without noticing that both intercepts coincide.

If both the x- and y-intercepts compute to (0,0), you only have ONE distinct point — you must find a second point by substituting any other x-value into the equation.

Summary

A line is fully determined by any two of its points.
Slope-intercept form: start at (0, b), then use rise/run from the slope to find a second point.
Intercept method: find where the line crosses each axis by setting the other variable to 0.
Always plot at least two points and draw the line through and beyond them.

References

WebsiteWikipedia — Linear equation