Transposing Terms in Equations
In the balance method, you add or subtract the same number from both sides. Transposition is a shortcut that gives the same result but is faster to write.
The rule is simple: when you move a term from one side of the equation to the other, change its sign. A positive number becomes negative, and a negative number becomes positive. Multiplication becomes division, and division becomes multiplication.
In Class 7 NCERT Maths, transposition is the most common method used to solve equations quickly.
What is Transposing Terms in Equations - Grade 7 Maths (Simple Equations)?
Definition: Transposition means moving a term from one side of an equation to the other side by changing its sign (or operation).
The rule:
- A term that is added on one side becomes subtracted on the other side.
- A term that is subtracted on one side becomes added on the other side.
- A term that multiplies on one side becomes a divisor on the other side.
- A term that divides on one side becomes a multiplier on the other side.
Transposing Terms in Equations Formula
Transposition Rules:
Change side, change sign (operation).
Summary table:
- x + a = b → x = b − a (+ becomes −)
- x − a = b → x = b + a (− becomes +)
- ax = b → x = b/a (× becomes ÷)
- x/a = b → x = b × a (÷ becomes ×)
Types and Properties
When to use transposition:
- Moving constants: When the variable is mixed with numbers on the same side, transpose the numbers to the other side.
- Moving variable terms: When the variable appears on both sides, transpose the variable terms to one side and constants to the other.
- Multi-step equations: First move additive/subtractive terms, then deal with multiplicative/divisive terms.
Important: Transposition is just a shortcut for the balance method. Both give the same answer. Transposition is quicker because you skip writing the intermediate step of performing the same operation on both sides.
Solved Examples
Example 1: Transposing an Added Term
Problem: Solve: x + 8 = 20
Solution:
Transpose +8 to RHS (it becomes −8):
x = 20 − 8 = 12
Check: 12 + 8 = 20. Correct!
Answer: x = 12
Example 2: Transposing a Subtracted Term
Problem: Solve: x − 15 = 7
Solution:
Transpose −15 to RHS (it becomes +15):
x = 7 + 15 = 22
Check: 22 − 15 = 7. Correct!
Answer: x = 22
Example 3: Transposing a Coefficient
Problem: Solve: 4x = 36
Solution:
4 multiplies x on LHS. Transpose to RHS as division:
x = 36 / 4 = 9
Check: 4 × 9 = 36. Correct!
Answer: x = 9
Example 4: Transposing Division
Problem: Solve: x/6 = 5
Solution:
6 divides x on LHS. Transpose to RHS as multiplication:
x = 5 × 6 = 30
Check: 30/6 = 5. Correct!
Answer: x = 30
Example 5: Two-Step Transposition
Problem: Solve: 3x + 5 = 20
Solution:
Step 1: Transpose +5 to RHS: 3x = 20 − 5 = 15
Step 2: Transpose ×3 to RHS: x = 15 / 3 = 5
Check: 3(5) + 5 = 15 + 5 = 20. Correct!
Answer: x = 5
Example 6: Variable on Both Sides
Problem: Solve: 5x − 3 = 2x + 9
Solution:
Step 1: Transpose 2x to LHS (becomes −2x): 5x − 2x − 3 = 9
Step 2: Simplify: 3x − 3 = 9
Step 3: Transpose −3 to RHS (becomes +3): 3x = 9 + 3 = 12
Step 4: Transpose ×3: x = 12 / 3 = 4
Check: LHS = 5(4) − 3 = 17. RHS = 2(4) + 9 = 17. LHS = RHS. Correct!
Answer: x = 4
Example 7: Equation with Negative Answer
Problem: Solve: 2x + 14 = 6
Solution:
Step 1: Transpose +14: 2x = 6 − 14 = −8
Step 2: Transpose ×2: x = −8 / 2 = −4
Check: 2(−4) + 14 = −8 + 14 = 6. Correct!
Answer: x = −4
Example 8: Equation with Fraction Result
Problem: Solve: 4x − 1 = 10
Solution:
Step 1: Transpose −1: 4x = 10 + 1 = 11
Step 2: Transpose ×4: x = 11/4
Check: 4(11/4) − 1 = 11 − 1 = 10. Correct!
Answer: x = 11/4 (or 2.75)
Example 9: Equation with Brackets
Problem: Solve: 2(x + 3) = 16
Solution:
Step 1: Expand the bracket: 2x + 6 = 16
Step 2: Transpose +6: 2x = 16 − 6 = 10
Step 3: Transpose ×2: x = 10 / 2 = 5
Check: 2(5 + 3) = 2(8) = 16. Correct!
Answer: x = 5
Example 10: Multiple Transpositions
Problem: Solve: 7x + 4 = 3x + 24
Solution:
Step 1: Transpose 3x to LHS: 7x − 3x + 4 = 24
Step 2: Simplify: 4x + 4 = 24
Step 3: Transpose +4: 4x = 24 − 4 = 20
Step 4: Transpose ×4: x = 20 / 4 = 5
Check: LHS = 7(5) + 4 = 39. RHS = 3(5) + 24 = 39. Correct!
Answer: x = 5
Real-World Applications
Why transposition is useful:
- Speed: Transposition is faster than writing out the full balance method, especially in exams.
- Multi-step equations: For equations with several terms, transposition keeps the working neat and organised.
- Higher classes: Transposition is used in Class 8, 9, and 10 for solving linear equations, quadratic equations, and algebraic identities.
- Science formulas: Rearranging formulas like v = u + at to find t requires transposition.
Key Points to Remember
- Change side, change sign is the rule of transposition.
- + becomes − and − becomes + when transposing.
- × becomes ÷ and ÷ becomes × when transposing.
- Transposition is a shortcut for the balance method.
- Always bring variable terms to one side and constant terms to the other.
- Deal with addition/subtraction first, then multiplication/division.
- Always check your answer by substituting back.
- Transposition works for all linear equations.
Practice Problems
- Solve by transposition: x + 13 = 25
- Solve: x − 9 = −4
- Solve: 6x = 42
- Solve: 5x + 3 = 28
- Solve: 4x − 7 = 2x + 5
- Solve: 3(x − 2) = 18
- Solve: x/3 + 5 = 11
Frequently Asked Questions
Q1. What is transposition in maths?
Transposition means moving a term from one side of an equation to the other by changing its sign or operation. For example, in x + 5 = 12, transpose +5 to get x = 12 − 5 = 7.
Q2. Why does the sign change when transposing?
Transposition is a shortcut for the balance method. When you subtract 5 from both sides of x + 5 = 12, you get x = 12 − 5. Instead of writing the subtraction step, you just move +5 as −5. The sign change represents the inverse operation.
Q3. Is transposition the same as the balance method?
Both give the same result. Transposition is a faster way to write what the balance method does. In the balance method, you write the operation on both sides. In transposition, you just move the term and change the sign.
Q4. Can you transpose the variable?
Yes. If the variable appears on both sides, transpose the variable term to one side. For example, in 5x = 2x + 9, transpose 2x to get 5x − 2x = 9, i.e., 3x = 9.
Q5. What if the answer is a fraction?
That is perfectly fine. Not all equations have whole number answers. For example, 3x = 10 gives x = 10/3. Leave it as a fraction or convert to a decimal.
Q6. Do you always transpose addition/subtraction first?
Yes. First transpose the terms being added or subtracted (to separate the variable term from constants). Then transpose the coefficient (multiplication or division) to find the value of the variable.
