In short
Transposition is the shorthand technique your CBSE Class 7 and 8 textbooks lean on when solving linear equations: you "move a term across the equals sign" and its sign flips mid-flight. A +5 on the left becomes a -5 on the right; a 5x on the right becomes -5x on the left. The underlying truth is the boring one — you are just subtracting the same thing from both sides. The slide-and-flip is bookkeeping, not magic. Once you see why the sign must flip (you are applying the inverse operation), transposition stops being a rule to memorise and becomes a rule you can re-derive in five seconds.
You are solving 2x + 5 = 11 for the hundredth time, and your textbook does this:
Notice the sneaky middle step. The +5 that was on the left is suddenly a -5 on the right. Nobody wrote a justification. Nobody wrote "subtracting 5 from both sides." The 5 just slid across the equals sign, and on the way it flipped its sign.
This move is called transposition, and it is the single most-used technique in Class 7 and 8 algebra. Your teacher probably said something like "jab term equals ke us paar jaata hai, sign badal jaata hai" — when a term goes to the other side of the equals sign, its sign changes. It works. You have used it a thousand times. But have you ever asked why the sign flips? It is not because the equals sign is a magical mirror. It is because transposition is shorthand for something much more basic — and once you see what it is shorthand for, the rule becomes obvious.
Watch the term slide
Click the button. Watch the +5 box detach itself from the left side, glide across the equals sign, and arrive on the right side as a -5. The flip happens mid-flight, right as the box crosses the bar of the =.
Click "Transpose 5". The orange $+5$ box slides right, crosses the equals sign, and arrives on the right side as $-5$. The new equation is $2x = 11 - 5 = 6$, so $x = 3$.
That is the whole picture. A term that was being added on one side becomes a term being subtracted on the other side. (And similarly: a term being subtracted becomes added; a factor being multiplied becomes a divisor; a divisor becomes a multiplier.) The sign or operation flips because you are undoing it on the side it left.
What is really going on (the boring truth)
Here is the move written out the long way:
On the left, +5 - 5 cancels to 0, so the +5 vanishes. On the right, 11 - 5 = 6, so a -5 appears. Why: the equals sign is a balance. If you take 5 off the left pan, you must take 5 off the right pan or the scale tips. That is the whole "do the same to both sides" rule, and it is the only rule of equation-solving you ever truly need.
Now look at what transposition records. You did not write the line where -5 appears on both sides. You skipped straight to the result. The +5 "moved" to the right side as -5, because that is exactly the net effect of subtracting 5 from both sides. Why the sign flips: you are applying the inverse operation. The +5 on the left was being added; to remove it you must subtract. The same subtraction, applied to the right side, is what shows up there as -5. The flip is not the equals sign performing magic — it is the inverse operation announcing itself.
So transposition is a notational shortcut for the inverse-operation step. Nothing more, nothing less. Every transposition can be re-written as a "do the same to both sides" move, and every "do the same to both sides" move that cancels a single term can be written as a transposition.
Three snapshots of the slide
The animation above is just three frozen moments stitched together. Here they are as static SVG, side by side:
Three snapshots of one transposition. The orange box is the term being moved; the green box is where it lands after the flip. The $\pm$ in the middle frame is the moment of the flip — the term is neither here nor there.
Worked examples
Transposing a constant
Solve 3x + 7 = 22.
The +7 is the term being added on the left. Transpose it across the equals sign — it lands on the right as -7:
Now divide both sides by 3 (or, equivalently, transpose the \times 3 as a \div 3):
Check. 3(5) + 7 = 15 + 7 = 22. ✓
The long-way version: subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Same arithmetic, two extra lines.
Transposing a variable term
Solve 2x = 5x - 9.
The variable x appears on both sides. To collect x-terms on the left, transpose the 5x from the right (where it is being added) to the left (where it lands as -5x):
Combine like terms on the left:
Now divide both sides by -3:
Check. Left side: 2(3) = 6. Right side: 5(3) - 9 = 15 - 9 = 6. ✓
Notice the same flip rule. The 5x was being added on the right, so it arrives as a subtraction on the left. The sign of any moved term flips, regardless of whether the term is a number or a multiple of x.
The trap — forgetting to flip
Solve 4x + 6 = 18. A student in a hurry writes:
They moved the +6 across the equals sign but forgot to flip its sign — they kept it as +6 on the right. The result is wrong: plug x = 6 back in and the left side is 4(6) + 6 = 30, not 18.
The correct move flips the sign:
Why is the wrong move wrong in the underlying-truth language? The student secretly added 6 to the right side instead of subtracting it — they did opposite operations to the two sides. The scale tipped. The shortcut "transpose and flip" packages the inverse operation into one move; skipping the flip is the same as skipping the inverse and just deleting the term, which is not a legal step.
When transposition is the wrong shortcut
Transposition is for terms being added or subtracted, and (with the same idea) for factors being multiplied or divided through the whole side. It does not apply to terms inside brackets, terms underneath a square root, or terms in a denominator paired with other terms. Take \frac{x + 3}{2} = 5. You cannot transpose the +3 alone, because it is locked inside the numerator. You must first multiply both sides by 2 to get x + 3 = 10, and then transpose the +3 to get x = 7. Why: the bracket or the fraction bar groups things together. The +3 is not "being added to the rest of the equation" — it is being added inside the numerator, then the whole numerator is being divided by 2. You must undo operations from the outside in.
This is why students who learn transposition as a magic rule sometimes write nonsense like \frac{x + 3}{2} = 5 \implies x = 5 \cdot 2 - 3. That happens to give the right answer here, but only by luck. Try \frac{x + 3}{2} = \frac{x}{4} with the same trick and you get garbage.
Why CBSE leans so hard on this
Open any NCERT Class 7 or Class 8 Maths textbook and you will see transposition introduced almost immediately after the chapter on equations begins. The reason is pedagogical: writing the long-way version every time — "subtract 5 from both sides, then…" — turns a three-line solution into a fifteen-line one, and beginning students lose the thread. Transposition compresses each inverse-operation step into a single visible move, so the structure of the solution stays clear. By the time you are solving simultaneous equations or word problems with three unknowns, you cannot afford to write out the long version of every move; transposition is the speed-up that makes the rest of school algebra readable.
But — and this is the catch — the speed-up only works if you understand what you are speeding up. A student who only knows "the sign changes when it crosses the equals sign" is one careless minute away from forgetting the flip and getting the wrong answer. A student who knows transposition is shorthand for "subtract from both sides" can re-derive the rule on the fly and never has to memorise it.
References
- NCERT Class 8 Mathematics, Chapter 2: Linear Equations in One Variable. [ncert.nic.in]
- Wikipedia, Equation solving — overview of the inverse-operation principle that transposition packages.
- Wikipedia, Equality (mathematics) — what the equals sign actually means.
- Khan Academy, Why we do the same thing to both sides of an equation — the underlying-truth version of transposition.