Systems of Equations

A "system of equations" is just two equations sharing the same two variables. The SAT almost always gives you a system where one of two classic methods — substitution or elimination — gets you to the answer in a few lines.

Elimination works best when the equations are already lined up (same variables, similar coefficients). Add or subtract the two equations so that one variable cancels out, leaving a single-variable equation to solve.

Substitution works best when one equation is already solved for a variable (e.g. y = 2x + 1). Plug that expression into the other equation in place of that variable, then solve.

Once you have one variable, plug it back into either original equation to find the other.

Example (elimination): x + y = 10 and x - y = 4. Add the equations: 2x = 14, so x = 7. Substitute back: 7 + y = 10, so y = 3.

Example (substitution): y = 3x and 2x + y = 20. Substitute y: 2x + 3x = 20, so 5x = 20, x = 4. Then y = 3(4) = 12.

Forgetting to solve for BOTH variables — the SAT usually asks for one specific value, but a careless reader stops after finding the first variable and picks the wrong answer choice. Also watch the sign when subtracting one whole equation from another; distribute the negative to every term.