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, 4x + 2y = 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 gives
-8 = (-2)(6) + 4
-8 = -12 + 4
-8 = -8
Thus the graph is correct.