Multi-Step Equations

The reliable order for any linear equation:

1. Distribute to clear parentheses.
2. Combine like terms on each side separately.
3. Move variables to one side, numbers to the other.
4. Divide by the coefficient.

For 3(x + 4) − 2x = 18:
3x + 12 − 2x = 18 → x + 12 = 18 → x = 6.

Check by substituting: 3(10) − 12 = 18. ✓

Checking is not optional politeness. It is a complete verification, and it catches sign errors that are otherwise invisible.

Every step must produce an equivalent equation — one with the same solution set. Adding the same expression to both sides and multiplying both sides by a nonzero constant preserve equivalence; multiplying by an expression that could be zero does not.

The standard order is chosen to minimise fractions and sign errors, but it is not the only valid order. In 3(x + 4) = 18, dividing by 3 first is faster than distributing. Recognising which operation was applied *last* to the variable side tells you which to undo first.

A linear equation in one variable, once simplified to ax = b, has exactly one solution when a ≠ 0, no solution when a = 0 and b ≠ 0 (an inconsistent equation such as 0 = 5), and infinitely many when a = 0 and b = 0 (an identity such as 0 = 0). Those two degenerate cases are worth recognising immediately: seeing the variables vanish is information, not an error.