Move a Term Across the Equals Sign and Its Sign Flips
Learn to tell an equation from an expression, solve by balancing and by transposition, handle brackets and unknowns on both sides, and clear fractional or decimal coefficients.
Why does a term change sign when it crosses the equals sign?
Because moving it across is a shorthand for doing the opposite operation to both sides. In , adding 4 to both sides gives — and that is exactly what "move the over as " describes.
Transposition is not a separate rule, just a faster way of writing the balancing step. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: equations versus expressions, balancing and transposition, brackets and unknowns on both sides, and fractional coefficients.
Transposition is not a separate rule, just a faster way of writing the balancing step. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: equations versus expressions, balancing and transposition, brackets and unknowns on both sides, and fractional coefficients.
What is the difference between an equation and an expression?
An expression has no equals sign and cannot be solved. An equation states that two quantities are equal, and it can be solved for the unknown.
So is an expression — it only has a value, which changes with . But is an equation, true for exactly one value of .
The value that makes it true is called the solution or root. To verify a value, substitute it into both sides separately and check they agree.
Test in :
Equal, so satisfies the equation.
Test :
Not equal, so is not a solution.
A weighing balance is the everyday picture — the two pans hold equal weights, which is what the equals sign asserts.
The habit worth building is checking LHS and RHS separately rather than working through the whole equation again. A verification question asks you to show both sides, and computing them side by side is the answer it wants.
So is an expression — it only has a value, which changes with . But is an equation, true for exactly one value of .
The value that makes it true is called the solution or root. To verify a value, substitute it into both sides separately and check they agree.
Test in :
Equal, so satisfies the equation.
Test :
Not equal, so is not a solution.
A weighing balance is the everyday picture — the two pans hold equal weights, which is what the equals sign asserts.
The habit worth building is checking LHS and RHS separately rather than working through the whole equation again. A verification question asks you to show both sides, and computing them side by side is the answer it wants.
How do the balancing and transposition methods work?
Balancing does the same operation to both sides. Transposition moves a term across the equals sign and changes its sign. They give identical working.
Solve by balancing:
The same by transposition:
The sign changes are:
- a term that was added becomes subtracted, and the reverse
- a factor that was multiplying becomes dividing, and the reverse
So in , the 4 was dividing, so it crosses as a multiplier: .
And in , the 5 was multiplying, so it crosses as a divisor: .
Always verify the answer in the original equation: . Correct.
One restriction applies to the division step: you may divide both sides by any number except zero, since division by zero is undefined.
Solve by balancing:
The same by transposition:
The sign changes are:
- a term that was added becomes subtracted, and the reverse
- a factor that was multiplying becomes dividing, and the reverse
So in , the 4 was dividing, so it crosses as a multiplier: .
And in , the 5 was multiplying, so it crosses as a divisor: .
Always verify the answer in the original equation: . Correct.
One restriction applies to the division step: you may divide both sides by any number except zero, since division by zero is undefined.
How do you solve equations with brackets or the unknown on both sides?
Expand the brackets first, then collect the unknowns on one side and the numbers on the other.
With the unknown on both sides. Solve . Transpose to the left and to the right:
Check: LHS and RHS . Equal.
With brackets. Solve :
With brackets on both sides. Solve :
Check: LHS and RHS . Equal.
The multiplier must reach every term inside the bracket. Writing as is the standard slip, and it produces a different answer that the verification step catches immediately.
When both brackets and split unknowns appear, the order never changes: expand, collect unknowns, collect numbers, divide.
With the unknown on both sides. Solve . Transpose to the left and to the right:
Check: LHS and RHS . Equal.
With brackets. Solve :
With brackets on both sides. Solve :
Check: LHS and RHS . Equal.
The multiplier must reach every term inside the bracket. Writing as is the standard slip, and it produces a different answer that the verification step catches immediately.
When both brackets and split unknowns appear, the order never changes: expand, collect unknowns, collect numbers, divide.
How do you clear fractional or decimal coefficients?
Multiply every term by the LCM of the denominators, which removes the fractions in one step.
Solve . The LCM of 2 and 3 is 6, so multiply throughout by 6:
Check: . Correct.
When each side is a single fraction, use cross-multiplication instead. Solve :
Check: LHS and RHS . Equal.
For decimal coefficients, multiply by a power of 10 or work directly. Solve :
Multiplying that one by 10 first gives , and the same .
The word every is what matters when clearing fractions. Multiplying only the fraction terms and forgetting the plain number on the right is the commonest error — in the first example the 5 had to become 30, not stay as 5.
Solve . The LCM of 2 and 3 is 6, so multiply throughout by 6:
Check: . Correct.
When each side is a single fraction, use cross-multiplication instead. Solve :
Check: LHS and RHS . Equal.
For decimal coefficients, multiply by a power of 10 or work directly. Solve :
Multiplying that one by 10 first gives , and the same .
The word every is what matters when clearing fractions. Multiplying only the fraction terms and forgetting the plain number on the right is the commonest error — in the first example the 5 had to become 30, not stay as 5.
Exam tip
Exam tip: verifying in the original equation
Equations are the one topic where you can prove your own answer right, and it takes two lines.
Substitute into the original equation — not your rearranged version, which may already carry the mistake — and compute LHS and RHS separately. If they match, the answer is correct.
Write one step per line with the operation shown, since method marks are awarded per step. A wrong final answer with correct working still scores.
When expanding a bracket, multiply every term inside. When clearing fractions, multiply every term by the LCM, including the constants.
And when a term crosses the equals sign, change its sign — and change the operation for a multiplier, so a multiplying 5 becomes a dividing 5.
Finally, answer what was asked. If the question wanted the number, is the answer; if it wanted a length, add the unit.
Substitute into the original equation — not your rearranged version, which may already carry the mistake — and compute LHS and RHS separately. If they match, the answer is correct.
Write one step per line with the operation shown, since method marks are awarded per step. A wrong final answer with correct working still scores.
When expanding a bracket, multiply every term inside. When clearing fractions, multiply every term by the LCM, including the constants.
And when a term crosses the equals sign, change its sign — and change the operation for a multiplier, so a multiplying 5 becomes a dividing 5.
Finally, answer what was asked. If the question wanted the number, is the answer; if it wanted a length, add the unit.
Did you know
Why is transposition just balancing written shorter?
Because crossing the equals sign is the opposite operation, performed on both sides at once.
Take . Balancing says: add 4 to the left, add 4 to the right. The left becomes — the has vanished — and the right becomes 15. Transposition simply skips writing the middle line and records the outcome: the reappears on the right as .
So the sign change is not a trick to memorise. It is the trace left behind by an operation you did to both sides, which is why every transposition step can be checked by asking what you actually added, subtracted, multiplied or divided.
Take . Balancing says: add 4 to the left, add 4 to the right. The left becomes — the has vanished — and the right becomes 15. Transposition simply skips writing the middle line and records the outcome: the reappears on the right as .
So the sign change is not a trick to memorise. It is the trace left behind by an operation you did to both sides, which is why every transposition step can be checked by asking what you actually added, subtracted, multiplied or divided.
Key takeaways
Simple linear equations: quick revision
- An expression has no equals sign and only a value; an equation asserts equality and has a solution.
- Verify a root by computing LHS and RHS separately: works in since both sides give 11.
- Balancing applies the same operation to both sides; transposition moves a term across and flips its sign, and a multiplying factor crosses as a divisor.
- Expand brackets fully — — then collect unknowns on one side and numbers on the other.
- For fractions, multiply every term by the LCM: becomes , so .
- With a single fraction on each side, cross-multiply: gives .
You will remember all of this far better after answering five questions on it than after reading it twice.
- Verify a root by computing LHS and RHS separately: works in since both sides give 11.
- Balancing applies the same operation to both sides; transposition moves a term across and flips its sign, and a multiplying factor crosses as a divisor.
- Expand brackets fully — — then collect unknowns on one side and numbers on the other.
- For fractions, multiply every term by the LCM: becomes , so .
- With a single fraction on each side, cross-multiply: gives .
You will remember all of this far better after answering five questions on it than after reading it twice.