Solving a Linear Equation

Solving a linear equation means undoing what was done to the variable — clearing parentheses, collecting like terms, and applying inverse operations to both sides until the variable stands alone.

Solve 3x + 7 = 19. Subtract 7 from both sides — the additive inverse at work: 3x = 12. Divide both sides by 3 — the multiplicative inverse: x = 4. Substitute back: 3(4) + 7 = 19 — true, so 4 is the solution. Every legal move applies the same operation to both sides, which is why the balance never tips: the two sides stay equal at every line, and the last line reads the answer. Longer equations add moves, never new kinds of moves.