How To Graph Linear Equations With Two Variables

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:

\(2y = -4x + 8\)

or

\(y = -2x + 4\)

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

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

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.

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

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.

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.

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

\(y = -2x + 4\)

gives

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

Thus the graph is correct.