elimination method

Picture a balance scale for each equation. If two scales each hold equal weights on both sides, you can pour one whole scale into the other and the result stays balanced. The elimination method exploits exactly this: you add or subtract entire equations so that one variable cancels out, leaving a simpler equation in the variable that remains.

The technique: line up the equations so like terms sit in columns, then multiply one or both equations by suitable numbers so that the coefficients of one variable become opposites (or equal). Adding (or subtracting) the equations then makes that variable vanish. Solve the resulting one-variable equation, and back-substitute to find the other.

Elimination is often the cleanest route when no variable is already isolated and the coefficients invite a quick cancellation. It is also the seed of Gaussian elimination, the systematic procedure used for larger systems. One honest reminder: you must scale a whole equation — every term, both sides — or you silently change the system and corrupt the answer.