How to Solve Systems of Linear Equations in Three Variables
Systems of linear equations will sometimes have three variables. These can be solved the same was as two-variable equations, but they take a few more steps. Linear equations with three variables will have an x, y and z variable. There are two ways to solve systems of linear equations with three variables, but the best way is by using the elimination method.
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Instructions
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1
Solve the three equations so that they are in the form x+y+z=4. By having the equations in the same order, you will be able to eliminate variables easier.
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2
Choose which variable you wish to eliminate. The coefficients of the variables must be exact opposites. You will be adding them together, so opposites are necessary for elimination. You will only be using two equations at a time.
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3
Add the two equations together to cancel out one of your variables. If nothing cancels out, you may need to multiply one or both of your equations by a number that will create equal, opposite coefficients. For instance, for the two equations x+2y+5z=1 and 3x+y+z=2, you will need to multiply the first one by -3 in order for the equations to cancel out.
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4
Save the resulting equation for later use.
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5
Using the third, or unused original equation, and either one of the other original equations, repeat Step 3. Be sure to cancel out the exact same variable as you did before.
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Solve one of your new equations for the variable of your choice. For instance, if you canceled out x in Step 3, you will solve for either y or z.
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Substitute the solved equation into the other new equation to solve for one of your variables.
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Substitute the solved variable into either of the new equations to solve for the second variable. At this point, you will have two of your variables solved.
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Substitute the two solved variables into any of your three original equations to solve for the third variable.
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Tips & Warnings
Check your results by substituting your answers into all three equations.
Always eliminate the same variable. Otherwise, you will not be able to create two equations with the two same variables.
