Solving systems by elimination

Standard form keeps columns honest

Put x-terms, y-terms, and constants in aligned columns. Missing terms can be written with coefficient zero.

Before adding, verify that the chosen variable has equal magnitude and opposite signs. Equal signs call for subtraction instead.

Multiply every term

If a coefficient needs scaling, multiply both sides and every term in the equation. Scaling only the convenient term changes the equation.

Choose small multipliers, often using the least common multiple of coefficient magnitudes.

Back-substitution and checking finish the work

Once one variable is found, substitute into whichever original equation has simpler coefficients. Then check the pair in both originals.

A zero row with a nonzero constant signals no solution; a zero row with zero signals dependent equations.

Worked example

Common mistakes

• Multiplying only one term when scaling an equation.
• Adding coefficients with the same sign and expecting cancellation.
• Forgetting to solve for the eliminated variable afterward.

Keep these ideas

• Align the system first.
• Create opposite coefficients with whole-equation moves.
• Check the final pair twice.