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

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

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:

or

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

gives

Thus the graph is correct.