How to Graph Linear Equations With Two Variables

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Graphs are among the most useful tools in mathematics for conveying information in a meaningful way. Even those who may not be mathematically inclined or have an outright aversion to numbers and computation can take solace in the basic elegance of a two-dimensional graph representing the relationship between a pair of variables.

Linear equations with two variables may appear in the form

Ax + By = C

and the resulting graph is always a straight line. More often, the equation takes the form

y = mx + b

where ​m​ is the slope of the line of the corresponding graph and ​b​ is its ​y​-intercept, the point at which the line meets the ​y​-axis.

For example, 4​x​ + 2​y​ = 8 is a linear equation since it conforms to the required structure. But for graphing and most other purposes, mathematicians write this as:

2y = -4x + 8


y = -2x + 4

The ​variables​ in this equation are ​x​ and ​y​, while the slope and ​y​-intercept are ​constants​.

Step 1: Identify the y-Intercept

Do this by solving the equation of interest for ​y​, if necessary, and identifying ​b​. In the above example, the ​y​-intercept is 4.

Step 2: Label the Axes

Use a scale convenient to your equation. You may encounter equations with unusually high of low values of the ​y​-intercept, such as −37 or 89. In these cases, each square of your graph paper might represent ten units rather than one, and so both the ​x​-axis and ​y​-axis should signify this.

Step 3: Plot the y-Intercept

Draw a dot on the ​y​-axis at the appropriate point. The y-intercept, incidentally, is simply the point at which ​x​ = 0.

Step 4: Determine the Slope

Look at the equation. The coefficient in front of ​x​ is the slope, which can be positive, negative, or zero (the latter in cases when the equation is just ​y​ = ​b​, a horizontal line). The slope is often called "rise over run" and is the number of unit changes in ​y​ for every single unit change in x. In the above example, the slope is −2.

Step 5: Draw a Line Through the y-Intercept with the Correct Slope

In the above example, starting at the point (0, 4), move two units in the ​negative​ ​y​-direction and one in the ​positive​ ​x​ direction, since the slope is −2. This leads to the point (1, 2). Draw a line through these points and extending in both directions for as far as you like.

Step 6: Verify the Graph

Pick a point on the graph distant from the origin and check to see if it satisfies the equation. For this example, the point (6, −8) lies on the graph. Plugging these values into the equation

y = -2x + 4


\begin{aligned} -8 &= (-2) × 6 + 4 \\ -8 &= -12 + 4 \\ -8 &= -8 \end{aligned}

Thus the graph is correct.


About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at