How to Solve Systems of Three Linear Equations by Elimination
One method of solving linear systems is elimination. This involves eliminating variables by adding two of the equations together. With variables eliminated, this makes solving for the remaining variables easier. Systems of three linear equations will have two or more variables you will need to solve for. The correct solution set will work for all three linear equations.
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Instructions
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1
Solve the equations so that they are in the same format. For instance, solve your set so that they are in the form x+y=5 or x+y+z=5. This will make it easier to add the equations together to eliminate a variable.
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2
Determine if a variable can be eliminated. The terms must be exact opposites in order to cancel out. You will only use two systems at a time. If no terms cancel out, you will need to multiply one or both of your chosen equations by a number that will create opposite coefficients. For instance, if using the two equations 2x+5y-4z=10 and x+3y+2z=5, you will need to multiply the second equation by -2 in order for the x variable to cancel out.
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Save the resulting equation from Step 2 for later use.
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Repeat Step 2 using your unused equation. You can use either of the other two equations with it. Cancel out the same variable that you did in Step 2. For instance, if you canceled out y, be sure you cancel out y again.
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5
Write down your two new equations and solve the simplest one for your chosen variable.
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Substitute the solved equation into the remaining equation and solve. This will give you the answer to one of your variables.
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Substitute the solved variable into either one of the new equations to solve for the second variable.
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Substitute both solved variables into one of the original equations to solve for the third variable, if you have one.
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Tips & Warnings
You can choose to eliminate any variable as long as you are consistent.
Always solve your equations so the variables are in the same order before you try to eliminate anything.
If you need to multiply to get a common coefficient, be certain to multiply not only the equation but the answer as well. For instance, if multiplying 2x+y=1 by 3, the result should be 6x+3y=3.
