Substitution

When one equation already says what a variable equals, push that expression into the other equation. In y = 2x - 1; x + y = 5 the first line hands over y, so the second becomes one equation in one variable:

x + (2x - 1) = 5

3x - 1 = 5

x = 2

Then y comes from the line that named it: y = 2(2) - 1 = 3. The solution is the pair x = 2, y = 3, the crossing computed instead of read off a picture.

Substitution shines exactly when a variable stands alone somewhere; the whole first move costs nothing because the isolating is already done.