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
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.