Solving One-Step Equations: notes and common mistakes

Key ideas

• An equation is a balance. A solution is a value of the variable that makes it true.
• To solve a one-step equation, do the inverse operation to both sides: addition and subtraction undo each other, and so do multiplication and division.
• Multiply or divide both sides only by numbers that are not zero.
• Always check by substituting the answer into the original equation.

Steps

1. Spot the one operation attached to the variable (add, subtract, multiply or divide).
2. Do the opposite operation to both sides.
3. Make sure the variable is alone on one side. If it ended up on the right, flip the equation.
4. Check by substituting into the original equation.

Formulas

• Add form: x + a = b gives x = b − a
• Subtract form: x − a = b gives x = b + a
• Multiply form: ax = b gives x = b ÷ a (when a ≠ 0)
• Divide form: x/a = b gives x = a × b (when a ≠ 0)
• Fraction coefficient: (p/q)x = b gives x = b × (q/p) (when p ≠ 0 and q ≠ 0)

Watch out

• Do the move to both sides, not just one.
• Use the opposite operation. In x − 9 = 4 you add 9; you do not subtract again.
• A lone minus sign is −1: in −x = 12 the answer is x = −12, not 12.
• Mind the signs when you divide: a negative divided by a negative is positive, and a positive divided by a negative is negative.
• Check in the original equation, not in a step you have already changed.

Why it works

• Why do the same thing to both sides? Equal things stay equal when you treat them equally.
• Why not multiply by 0? Both sides become 0, and 0 = 0 is true no matter what x is, so you lose track of the solution. Dividing by 0 is not possible.
• Why use the opposite operation? It cancels what is stuck to x: subtracting 7 cancels adding 7, and multiplying by 4/3 cancels multiplying by 3/4.