Systems of Equations

Two equations, two unknowns — by substitution and elimination.

The idea

A system of equations imposes two equations at once on the same unknowns, and a solution is a pair $(x, y)$ that satisfies both together. One linear equation in two unknowns allows infinitely many pairs, one for every point on a line, so the solution of the system is the point where the two lines cross.

Two methods find that pair, and both work by removing one unknown so that an ordinary one-variable equation is left.

Elimination adds or subtracts the two equations, which is legal because equals added to equals stay equal. When the coefficients of one variable are opposites, that variable vanishes from the sum; when they are not, first multiply a whole equation through by a constant to make them so.

Substitution solves one equation for one variable and puts the resulting expression wherever that variable appears in the other equation. It is the natural choice when an equation already isolates a variable.

Either method yields one coordinate. Substitute it into either original equation to recover the other, and check the pair against both.

Ways to work on it

• Walkthrough. Elimination and substitution on two small systems.
• Practice. Solve a random 2 × 2 system for x and y.
• Hardest. Elimination that needs a multiplier to align coefficients.

Not sure where to start? Take the ten-question placement test.