How to Solve Systems of Three Linear Equations

Linear systems are groups of equations with x and y variables. These variables correspond to the x-axis and y-axis on a graph. Some systems have three equations. These take a little more work, but are no harder to solve than systems with two equations. All linear systems can have their solutions graphed by using the solved x, y coordinates.

Things You'll Need

• Graph paper

Instructions

1. Write down your system.This article will use the following system of equations:x+y=92x+4y=10x-2y=21

2. Solve the simplest equation for y. Some systems may have one equation already solved. For instance, you may have x=0, or y=-1. If this is the case, proceed to Step 3 without solving any of the other equations.The simplest equation in this case is x+y=9. Solving this for y yields y=9-x.

3. Substitute your equation from Step 2 in place of the y variable in one of the remaining equations. You can use either equation. Choose the equation that will be easier to solve.Substitute y=9-x for y in x-2y=21.x-2(9-x)=21x-18+2x=213x-18=213x=39x=13

4. Substitute the value of x you found in Step 3 into any of the three equations to find the value of y.2x+4y=102(13)+4y=1026+4y=104y=-16y = -4

5. Check your answer by placing the x and y answers into each equation. If the equation isn’t correct, repeat Steps 2 through 4.Check x+y=9. 13+-4=9 9=9Check 2x+4y=10. 2(13)+4(-4)=1026+-16=1010=10Check x-2y=21.13-2(-4)=2113+8=21

Tips & Warnings

• Graph your solution by setting up a table of values for each equation. All lines should connect at the point found in Step 4.

• Always check your results. The solution set should work for all three equations; otherwise, it’s incorrect.

Resources

• Ways to Solve Linear Systems