What Are Simultaneous Equations?

One equation with two unknowns, like $x + y = 7$, has many solutions: $(1,6)$, $(2,5)$, $(3,4)$… A single equation cannot pin down two unknowns.

Add a second condition — say $x - y = 1$ — and suddenly only one pair satisfies both at once: $x = 4, y = 3$. Two equations that must hold at the same time are called simultaneous equations, and a solution is a pair $(x, y)$ that makes both true.

Checking a pair works by double substitution: test the pair in equation 1 AND equation 2. Passing one test is not enough.