Systems of Equations: Substitution and Elimination

Each linear equation describes a line made of every (x, y) pair that satisfies it. If two lines are different and not parallel, they cross at exactly one point, and that point's coordinates are the only numbers that satisfy both equations at once. Solving a system means finding that crossing point without graphing.

Substitution does it by shrinking two unknowns into one. Solve one equation for one variable, then plug that expression into the other equation. You now have a single equation in a single variable, which you can solve normally, then plug back in to get the second value. Elimination works by adding or subtracting the two equations so one variable cancels. If the coefficients on a variable aren't already opposites, multiply one or both equations by numbers that make them match, then add. Either method turns two unknowns into one unknown, one step at a time.

These two moves sit underneath almost everything else in this unit: quadratic systems, rational equations cleared of denominators, even matrix methods all reduce to the same idea of eliminating variables until one is left standing. The habit to build now is reading exactly what's being asked. A system gives you x and y separately, but a question might want x + y, x times y, or 2x - y. Solve the system fully, then build whatever combination the question actually wants from both values.