Simple Simultaneous Equations

Simultaneous equations are two equations with two unknowns (usually $x$ and $y$) that must both be true at the same time. The solution is the single pair of values that satisfies both.

Method 1 — Elimination. Add or subtract the equations to cancel one variable.

Solve $x + y = 10$ and $x - y = 4$. Add them: the $y$'s cancel, giving $2x = 14$, so $x = 7$. Put $x = 7$ into the first: $7 + y = 10$, so $y = 3$.

Method 2 — Substitution. If one equation gives a variable on its own, substitute it into the other.

Solve $y = x + 1$ and $2x + y = 7$. Replace $y$: $2x + (x + 1) = 7$, so $3x + 1 = 7$, giving $x = 2$ and then $y = 3$.