How to Solve Equations by Graphing

Using a graphing calculator can give you solutions to complex equations.the solution for the equation.

Solving a system of equations is a multi-step algebraic process, involving substitution and manipulating multiple variables. Sometimes, finding a solution may seem impossible (especially if the equations are complex and involve radicals, square roots and other complicated expressions). Graphing those same equations can provide you with a solution: the intersection of the graphs will be the solution for the equation. For example, the solution to the system of linear equations y= 5(x - 4) and y = 3x + 2 can be found by graphing the two on the same X,Y axis.

Things You'll Need

• Graph paper
• Pen
• Pencil

Instructions

1. Figure out two points on each equation by solving. For example, two points on the graph of the equation y= 5(x - 4) would be (for x=1):
   y = 5((1)-4) = -15, and
   y = 5((10-4) = 30
   The values you choose for x should be kept low, but spread the points apart a little. For example, solve for x=1 and x=10. You could draw a table of values and use that too, but that would be a lot more work (linear equations are a line, so you only need two points on the line to graph it).

2. Draw an X,Y axis on your paper by drawing a horizontal line across the center of the paper and a vertical line down the center of the paper. It should look like a big cross.

3. Label your X and Y axis. The labels will depend on the coordinates you solved for in Step 1. For example, if your coordinate (x,y) pairs are (1,-15) and (10,20) you should label your X axis from -20 to -20 and your Y axis from -20 to 20.

4. Plot your first coordinate pair. For example, to plot the coordinate pair 1, -15, find "1" on the X axis then go down 15 spaces on the Y axis to Y=-15. Make a small dot at that place and label it (1, -15). .

5. Plot the remaining coordinate for your first equation.

6. Join the two dots together with a ruler and pencil, making a straight line.

7. Graph the second equation, following the guidelines in Step 4 to Step 6.

8. Find the intersection of the two lines (the place where the two lines cross). This will be the solution to your equation. For example, if your graph crosses at the intersection of (4,10) then that is the solution to the system of equations (x=4, y=10).

Tips & Warnings

• Try graphing the equations on a graphing calculator. It's much faster than graphing by hand and you can get an exact answer to as many decimal places as you need.

• Don't use this method for quadratic equations (or other curvy equations). It's impossible to get an exact answer because the graph isn't straight.

References

• Teacher's Choice: Solve Equations by Graphing