Solving Two-Step Equations

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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.

2x + 5 = 13 x was ×2, then +5 −5 both sides FIRST ÷2 both sides
Undo in reverse: the +5 was applied last, so it comes off first. Addition/subtraction always unwraps before multiplication/division.

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.

START 2x+5 13 −5 both REMOVE 5 EACH SIDE 2x 8 ÷2 both SPLIT IN HALF x 4 2x + 5 = 13 → 2x = 8 → x = 4 Remove 5 from each side, then split each side in half — x = 4. Teal blocks are x-blocks; gold blocks are units.
The golden rule of equations: do the same thing to both pans and the scale stays balanced. Remove 5 from each side, then split each side in half — x = 4.
3x − 7 = 8 +7 both 3x = 15 ÷3 both x = 5 The −7 was applied last, so it comes off first.
Undo in reverse: the −7 was applied last, so it comes off first. Addition/subtraction always unwraps before multiplication/division.

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.

Example 1 Basic: 2x + 5 = 13
  1. Thinking: x is multiplied by 2, then 5 is added. Undo in reverse: subtract 5 first, then divide by 2.
  2. Subtract 5 from both sides: 2x = 13 − 5 = 8.
  3. Divide both sides by 2: x = 8 ÷ 2 = 4.
  4. Check: 2(4) + 5 = 8 + 5 = 13 ✓.
x = 4
Example 2 Subtraction inside: 3x − 7 = 8
  1. Thinking: the last operation applied to x was “subtract 7,” so add 7 to both sides first.
  2. Add 7 to both sides: 3x = 8 + 7 = 15.
  3. Divide both sides by 3: x = 15 ÷ 3 = 5.
  4. Check: 3(5) − 7 = 15 − 7 = 8 ✓.
x = 5
Example 3 Division: x/4 + 2 = 9
  1. Thinking: x was divided by 4, then 2 was added. Reverse: subtract 2, then multiply by 4.
  2. Subtract 2 from both sides: x/4 = 9 − 2 = 7.
  3. Multiply both sides by 4: x = 7 × 4 = 28.
  4. Check: 28/4 + 2 = 7 + 2 = 9 ✓.
x = 28
Example 4 Tricky signs: 5 − 2x = 11
  1. Thinking: careful — this is 5 minus 2x. Subtract 5 from both sides first.
  2. Subtract 5 from both sides: −2x = 11 − 5 = 6.
  3. Divide both sides by −2: x = 6 ÷ (−2) = −3. (Dividing by a negative flips the sign of the answer — nothing else changes.)
  4. Check: 5 − 2(−3) = 5 + 6 = 11 ✓. The negative solution is correct — always verify with substitution.
x = −3

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.

Mistake 1: Undoing in the wrong order
Wrong
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.
Right
2x + 5 = 13 → 2x = 8 → x = 4
The +5 was applied last, so it comes off first.
Rule: shoes before socks — undo the operations in reverse order. Addition/subtraction unwraps before multiplication/division.
Mistake 2: Sign errors when moving terms
Wrong
5 − 2x = 11 → 2x = 6 → x = 3
The −2x magically became +2x. Terms do not change sign on their own.
Right
5 − 2x = 11 → −2x = 6 → x = −3
Check: 5 − 2(−3) = 11 ✓.
Rule: terms carry their signs with them like name tags. −2x stays −2x until you divide — it never flips on its own.
Mistake 3: Skipping the check
Wrong
Solving 7 − 3x = 1 and answering x = −2 without verifying.
(The check would have caught it: 7 − 3(−2) = 13, not 1.)
Right
Always substitute back. If x = 2: 7 − 3(2) = 1 ✓.
Ten seconds of checking catches nearly every arithmetic slip.
Rule: the check takes ten seconds and catches nearly every arithmetic slip. Professionals check; amateurs hope.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Solve 4x + 3 = 19.
Answer
4x = 19 − 3 = 16, so x = 16 ÷ 4 = 4.
Check: 4(4) + 3 = 19 ✓.
2. Solve 5x − 3 = 27.
Answer
5x = 27 + 3 = 30, so x = 30 ÷ 5 = 6.
Check: 5(6) − 3 = 27 ✓.
3. Solve x/3 + 4 = 10.
Answer
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.
Practice two-step equations Try the equation tool Explore the balance lab
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