Transposition

Transposition Method

What you'll learn

Solve equations quickly by moving terms from one side of the equals sign to the other.
Apply the transposition rule: a term changes side and sign.
Use transposition for equations with variables on both sides.
Connect transposition to the balanced-equation rule you already know.

Key concepts

Level 1 — Core idea

What is transposition?
Transposition is a shortcut for solving equations. Instead of adding/subtracting on both sides, you move a term to the opposite side and reverse its sign.
The transposition rule
- +a on LHS moves to RHS as −a.
- −a on LHS moves to RHS as +a.
- ×a (as coefficient) moves to RHS as ÷a.
- ÷a moves to RHS as ×a.
  Example: x + 5 = 12 → x = 12 − 5 (the +5 became −5 on the other side).
Why it works — Moving +5 from LHS to RHS is exactly the same as subtracting 5 from both sides:
x + 5 − 5 = 12 − 5 → x = 12 − 5.

Level 2 — Process and representation

Variables on both sides — Collect variable terms on one side and constants on the other using transposition.
Example: 5x − 3 = 2x + 9 → 5x − 2x = 9 + 3 → 3x = 12 → x = 4.
Order of transposition — There is no fixed order, but a useful strategy:
1. Move all variable terms to one side.
2. Move all constant terms to the other side.
3. Divide (or multiply) to isolate the variable.
Brackets — Expand or simplify brackets before transposing if needed.

Worked example

Solve by transposition: x − 7 = 18

Step 1 — transpose −7 to RHS; sign becomes +7:
         x = 18 + 7
Step 2 — x = 25
Verify: 25 − 7 = 18 ✓

Solve: 4y + 9 = 2y + 21

Step 1 — transpose 2y to LHS: 4y − 2y + 9 = 21
Step 2 — transpose +9 to RHS: 2y = 21 − 9
Step 3 — 2y = 12 → y = 6
Verify: LHS = 4(6) + 9 = 33; RHS = 2(6) + 21 = 33 ✓
Answer: y = 6

Solve: (3n/4) − 2 = 7

Step 1 — transpose −2: 3n/4 = 7 + 2 = 9
Step 2 — transpose ÷4 (coefficient of n): 3n = 9 × 4 = 36
Step 3 — transpose ×3: n = 36/3 = 12
Verify: (3×12/4) − 2 = 9 − 2 = 7 ✓
Answer: n = 12

Common mistakes

Misconception	What students think	Scientific correction
Transposing only part of a term: in 2x + 5 = 11, move *	Transposing only part of a term: in 2x + 5 = 11, move +5 as a whole, not just the 5 without changing sign.	Check the Key concepts and worked example for the NCERT-accurate version.
When moving 2x from RHS to LHS, write −2x, not	When moving 2x from RHS to LHS, write −2x, not +2x.	Check the Key concepts and worked example for the NCERT-accurate version.
Forgetting that transposition of multiplication/divisio	Forgetting that transposition of multiplication/division applies to the coefficient of the variable, not to constants being added.	Check the Key concepts and worked example for the NCERT-accurate version.

Quick check

Solve by transposition: m + 14 = 31. (m = 17)
Solve: 5x − 8 = 3x + 10. (x = 9)
Solve: p/3 + 4 = 9. (p = 15)
Explain in one line why x + 3 = 10 gives x = 10 − 3. (Subtracting 3 from both sides is the same as moving +3 to the other side as −3)

Open the Practice tab for graded questions on the transposition method.

Transposition

Transposition

Transposition Method

What you'll learn

Key concepts

Level 1 — Core idea

Level 2 — Process and representation

Worked example

Common mistakes

Quick check

Key Takeaways (TL;DR)

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