How to Interpret Linear Equations

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Simply put, a linear equation draws a straight line on a regular x-y graph. The equation holds two key pieces of information: the slope and the y-intercept. The slope’s sign tells you if the line rises or falls as you follow it left to right: A positive slope rises, and a negative one falls. The slope’s size governs how steeply it rises or falls. The intercept indicates where the line crosses the vertical y-axis. You’ll need beginning algebra skills to interpret linear equations.

Graphical Method

    Draw a vertical Y axis and horizontal X axis on the graph paper. The two lines should meet close to the center of the paper.

    Get the linear equation into the form Ax + By = C if it is not already in that form. For example, if you start with y = -2x + 3, add 2x to both sides of the equation to obtain 2x + y = 3.

    Set x = 0 and solve the equation for y. Using the example, y = 3.

    Set y = 0 and solve for x. From the example, 2x = 3, x = 3/2

    Plot the points you just obtained for x = 0 and y = 0. The example’s points are (0,3) and (3/2,0). Line the ruler up on the two points and connect them, passing the line through the x and y axis lines. For this line, note that it has a steep downward slope. It intercepts the y-axis at 3, so the has a positive beginning and proceeds downward.

Slope-Intercept Method

    Get the linear equation into the form y = Mx + B, where M equals the line's slope. For example, if you begin with 2y – 4x = 6, add 4x to both sides to obtain 2y = 4x + 6. Then divide through by 2 to get y = 2x + 3.

    Examine the equation’s slope, M, which is the number by x. In this example, M = 2. Because M is positive, the line will increase going left to right. If M were smaller than 1, the slope would be modest. Because the slope is 2, the slope is fairly steep.

    Examine the equation’s intercept, B. In this case, B = 3. If B = 0, the line passes through the origin, which is where the x and y coordinates meet. Because B = 3, you know that the line never passes through the origin; it has a positive beginning and steep upward slope, rising three units for every unit of horizontal length


    • Linear equations help you judge whether real-world tasks are successful. If the equation in the first example describes the results of your weight-loss regimen, you may be losing weight too rapidly, indicated by the steep downward slope. If the equation in the second example describes custom T-shirt sales, sales are increasing rapidly, and you may need to hire more help.

      A graphing calculator can rapidly draw graphs of linear equations, if you deal with them frequently.


About the Author

Chicago native John Papiewski has a physics degree and has been writing since 1991. He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. He also contributed to the book, "Nanotechnology: Molecular Speculations on Global Abundance." Please, no workplace calls/emails!

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  • Digital Vision./Digital Vision/Getty Images

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