Solving Multi-Step Equations: notes and common mistakes

Key ideas

• Multi-step equations take more than two moves. Tidy each side first (distribute, then combine like terms), then undo, then check.
• Tidying rewrites one side with the same value, so it is not a balance move. The undo moves must be done to both sides.
• For equations shaped like ax + b = c, undo add or subtract first, then multiply or divide.
• A number or a minus sign in front of parentheses multiplies every term inside, sign included.
• To clear fractions, multiply every term on both sides by a common denominator. To undo a fraction that multiplies a group, multiply both sides by its reciprocal.
• Check in the original equation.

Steps

1. If there are fractions, you can clear them first: multiply every term on both sides by a common denominator.
2. Distribute any number (with its sign) over every term inside parentheses.
3. Combine like terms on each side.
4. Undo the add or subtract on both sides.
5. Undo the multiply or divide on both sides (never by zero).
6. Check by substituting into the original equation.

Formulas

• Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac
• Minus before parentheses: −(b + c) = −b − c and −(b − c) = −b + c
• Reciprocal: (p/q) · (q/p) = 1 (when p ≠ 0 and q ≠ 0)

Watch out

• Distribute to every term: 3(x + 2) = 3x + 6, not 3x + 2.
• A negative times a negative is positive: −2(x − 5) = −2x + 10.
• A minus sign before parentheses flips every sign inside: 25 − (2x + 3) = 25 − 2x − 3.
• Only like terms combine: 3x + 5 cannot be added into 8x.
• When you clear fractions, multiply every term, including the whole numbers.
• Check in the original equation, not the tidied one.

Why it works

• Why can you tidy one side without touching the other? Distributing and combining only rewrite an expression as an equal expression. Its value does not change, so the balance is untouched.
• Why does distributing work? 3(x + 2) means 3 copies of (x + 2), which hold 3 copies of x and 3 copies of 2.
• Why check in the original equation? A slip made while tidying carries into the tidied equation. Only the original equation tells the truth about the problem.