Solving by Elimination

Solving Systems by Elimination

The elimination method (sometimes called the addition method) solves a system by adding or subtracting the equations to cancel out one of the variables entirely.

The Elimination Method

1. Line up both equations with like terms in columns.
2. If necessary, multiply one or both equations by a constant so that the coefficients of one variable become opposites (same size, opposite sign).
3. Add the two equations together — the chosen variable should cancel out.
4. Solve the resulting one-variable equation.
5. Substitute back into either original equation to find the other variable.
6. Check your ordered-pair solution in both original equations.

When to Multiply One Equation

If one variable's coefficients are already opposites or equal, you can add or subtract right away. If not, look for a coefficient in one equation that is a multiple of the matching coefficient in the other — multiplying just that one equation will line them up.

When to Multiply Both Equations

If neither coefficient is a multiple of the other, multiply both equations by suitable numbers so that a chosen variable's coefficients become equal in size (usually using the least common multiple of the two coefficients), then make them opposite in sign if needed by subtracting instead of adding, or by using a negative multiplier.

Example

Solve the system:

1. The y-coefficients, +2 and -2, are already opposites, so no multiplying is needed.
2. Add the two equations together so y cancels: (3x + x) + (2y - 2y) = 12 + 4 \;\Rightarrow\; 4x = 16
3. Solve for x: x = 4
4. Substitute x = 4 into the second original equation to find y: 4 - 2y = 4 \;\Rightarrow\; -2y = 0 \;\Rightarrow\; y = 0
5. The solution is (x, y) = (4, 0).

Check in both equations: 3(4)+2(0) = 12 and 4 - 2(0) = 4. Both are true.

Choosing Which Variable to Eliminate

When neither variable's coefficients line up neatly, pick whichever variable requires smaller multipliers. For example, in the system 2x + 3y = 16 and 5x - 2y = 11, the y-coefficients (3 and -2) have a least common multiple of 6, so multiplying the first equation by 2 and the second by 3 lines up the y-terms as +6y and -6y, ready to add and eliminate y.

Key Takeaways

• Elimination cancels a variable by adding or subtracting the two equations.
• Multiply one or both equations by constants until a variable's coefficients are opposites.
• After eliminating one variable, solve for the other, then substitute back for the first.
• Always check the ordered-pair solution in both original equations.