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Solving simultaneous linear equations by elimination

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Solving Systems by Elimination

The elimination method solves a system of linear equations by adding or subtracting the equations so one variable cancels, leaving a single equation to solve. Learn how to line up terms, multiply to match coefficients, eliminate a variable, and back-substitute, with a worked example.

What the elimination method is

The elimination method solves a system of linear equations by adding or subtracting the equations so that one variable cancels, leaving a single equation in one variable to solve.

The steps

The elimination method, step by step Worked example. The system is 2x plus 3y equals 12 and 2x minus y equals 4. The x coefficients already match, so subtracting the second equation from the first eliminates x and gives 4y equals 8, so y equals 2. Back-substituting y equals 2 into the first equation gives x equals 3. 2x + 3y = 122x − y = 4 x-coefficients match → subtract to eliminate x 4y = 8 y = 2 Back-substitute y = 2 into 2x + 3y = 12: 2x + 6 = 12 x = 3
Elimination: match a variable's coefficients, subtract to cancel it, solve, then back-substitute.

Line up the equations, multiply one or both so a variable has matching coefficients, add or subtract to eliminate that variable, solve for what remains, then back-substitute to find the other variable.

Worked example

Solve 2x + 3y = 12 and 2x − y = 4. The x-coefficients already match, so subtracting gives 4y = 8, so y = 2. Substituting into the first equation gives 2x + 6 = 12, so x = 3. The solution is (3, 2).

Elimination vs substitution

Elimination is quickest when coefficients already line up; otherwise substitution may be easier. Either way, a system can have one solution, none, or infinitely many — see determining the number of solutions.

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