Skill: two-step-equations
Solving Two-Step Equations
A two-step equation like 2x + 5 = 13 has two operations wrapped around x. Solving it means unwrapping x by undoing the operations in reverse order — and that one idea scales up to every equation you will ever meet.
1 Understand: Unwrap in Reverse
A two-step equation hides x under two operations. In 2x + 5 = 13, x was first multiplied by 2, then 5 was added. To free x, undo the operations in the opposite order they were applied.
The shoes-before-socks rule
You put on socks first, then shoes — but you take off the shoes first. Equations work the same way: the operation applied last comes off first.
- Addition/subtraction always unwraps before multiplication/division.
- Each unwrapping step must happen on both sides — that is what keeps the equation true.
- After two unwrappings, x stands alone: that is the solution.
This is the first equation type where strategy matters, not just arithmetic — and the strategy never changes, no matter how complicated equations get.
Where it is used
“I bought 3 notebooks and paid $2 shipping for $17 total — what did each notebook cost?” · converting temperatures · splitting a bill with a fixed fee. Any situation with a rate plus a fixed amount is a two-step equation.
2 See It: The Balance Scale
The golden rule of equations: do the same thing to both pans and the scale stays balanced.
3 Worked Examples
Follow each step. The pattern is always the same: undo the addition/subtraction first, then the multiplication/division — and check by substitution.
- Thinking: x is multiplied by 2, then 5 is added. Undo in reverse: subtract 5 first, then divide by 2.
- Subtract 5 from both sides: 2x = 13 − 5 = 8.
- Divide both sides by 2: x = 8 ÷ 2 = 4.
- Check: 2(4) + 5 = 8 + 5 = 13 ✓.
- Thinking: the last operation applied to x was “subtract 7,” so add 7 to both sides first.
- Add 7 to both sides: 3x = 8 + 7 = 15.
- Divide both sides by 3: x = 15 ÷ 3 = 5.
- Check: 3(5) − 7 = 15 − 7 = 8 ✓.
- Thinking: x was divided by 4, then 2 was added. Reverse: subtract 2, then multiply by 4.
- Subtract 2 from both sides: x/4 = 9 − 2 = 7.
- Multiply both sides by 4: x = 7 × 4 = 28.
- Check: 28/4 + 2 = 7 + 2 = 9 ✓.
- Thinking: careful — this is 5 minus 2x. Subtract 5 from both sides first.
- Subtract 5 from both sides: −2x = 11 − 5 = 6.
- Divide both sides by −2: x = 6 ÷ (−2) = −3. (Dividing by a negative flips the sign of the answer — nothing else changes.)
- Check: 5 − 2(−3) = 5 + 6 = 11 ✓. The negative solution is correct — always verify with substitution.
4 Common Mistakes
These three errors show up on almost every two-step-equations quiz. Spot them now and they will never cost you points.
2x + 5 = 13 → x + 5 = 6.5
Divided by 2 first — but the +5 is not attached to just one x, so you cannot split it yet.
2x + 5 = 13 → 2x = 8 → x = 4
The +5 was applied last, so it comes off first.
5 − 2x = 11 → 2x = 6 → x = 3
The −2x magically became +2x. Terms do not change sign on their own.
5 − 2x = 11 → −2x = 6 → x = −3
Check: 5 − 2(−3) = 11 ✓.
Solving 7 − 3x = 1 and answering x = −2 without verifying.
(The check would have caught it: 7 − 3(−2) = 13, not 1.)
Always substitute back. If x = 2: 7 − 3(2) = 1 ✓.
Ten seconds of checking catches nearly every arithmetic slip.
5 Quick Check
Try each one on paper first, then reveal the answer.
4x = 19 − 3 = 16, so x = 16 ÷ 4 = 4.
Check: 4(4) + 3 = 19 ✓.
5x = 27 + 3 = 30, so x = 30 ÷ 5 = 6.
Check: 5(6) − 3 = 27 ✓.
x/3 = 10 − 4 = 6, so x = 6 × 3 = 18.
Check: 18/3 + 4 = 10 ✓.
Key Points to Remember
- Unwrap x by undoing operations in reverse order: addition/subtraction first, then multiplication/division.
- Every unwrapping step happens on both sides — that is what keeps the equation true.
- Terms carry their signs like name tags: −2x stays −2x until you divide.
- Negative answers are fine — the substitution check is the final judge.
- Always check by substituting back into the original equation.