Flip the Sign for Added Terms, Invert for Multiplied Factors — Two Parallel Transposition Rules

You know the move. Your textbook writes x + 7 = 12 and the next line is x = 12 - 7. The +7 slid to the other side and its sign flipped. Then the very same textbook writes 3x = 15 and the next line is x = 15/3. This time the 3 slid to the other side and the operation flipped — \times 3 became \div 3. Same trick? Different trick? Your teacher probably did both moves in the same breath without flagging the difference.

They are two parallel rules with the same heart. Both are saying "apply the inverse operation to both sides." The catch is that there are two kinds of inverse — the additive inverse (the negative) and the multiplicative inverse (the reciprocal) — and which one you use depends on whether the term is glued to the rest by a + or by a \times. Once you see the pattern, you stop having to memorise either rule. You just ask: how is this thing attached?

The two rules, side by side

Here is the cheat sheet. Print it, paste it on your desk if you have to.

The shapes of the two rules are identical. Something crosses the equals sign and arrives as its inverse on the other side. The only thing that differs is which inverse — and that is dictated by whether the thing was glued on by addition or by multiplication.

Why the rules look different but really are the same

Both rules are shorthand for the do-the-same-to-both-sides law from Operations and Properties. The reason they look different is that the two arithmetic operations have two different inverses.

For addition, the inverse of "add c" is "subtract c". So when you want to peel off a +c from the left side of A + c = B, you subtract c from both sides:

Why: the additive inverse of any number c is -c. Adding c and then adding -c gives 0, so the original +c vanishes from the left. The right side keeps a record of what you did — a -c tacked onto B.

For multiplication, the inverse of "multiply by c" is "divide by c" (or, equivalently, multiply by 1/c). To peel off a \times c from the left side of A \cdot c = B, you divide both sides by c:

Why: the multiplicative inverse of any non-zero number c is its reciprocal 1/c. Multiplying by c and then by 1/c gives 1, so the original \times c vanishes from the left. The right side records the inverse — B divided by c. The condition c \neq 0 is essential, because 1/0 is undefined.

Transposition skips writing the cancellation step. Instead of subtracting (or dividing) on both sides explicitly, you just move the term across and apply the right inverse to it on its way over. That is the only difference between the textbook's three-line solution and the long-form six-line one.

Worked examples

Why the order matters when peeling — and why it doesn't on transposition

You may have noticed a pattern in the combined example: I peeled the additive layer (+7) first, then the multiplicative one (\times 3). That is the reverse of BODMAS / order of operations. When building the expression 3x + 7 from x, you would first multiply by 3, then add 7. When unbuilding — solving for x — you reverse the order: undo the addition first, then the multiplication. This is the standard "peel layers from the outside in" rule.

But here is a subtlety. Once you commit to transposition as the bookkeeping, the order is no longer forced — as long as each move is correct. Watch.

Take 3x + 7 = 22 again, but try peeling the \times 3 first:

This is also valid! What you did is divide both sides by 3. On the left, \frac{3x + 7}{3} = x + \frac{7}{3} (because division distributes over the sum). On the right, \frac{22}{3}. Now transpose the \frac{7}{3}:

Same answer. Why: transposition is just shorthand for inverse operations applied to both sides, and the do-the-same-to-both-sides rule is order-independent. You can divide first then subtract, or subtract first then divide. Both routes preserve the equation, so both arrive at the same solution.

So why do textbooks always do additive first? Cleaner arithmetic. If you divide first, fractions appear immediately and the next step gets messy. Subtracting first keeps the numbers integer for as long as possible. It is a habit, not a law.

Common confusions

• "The sign flips even for the multiplicative case." No. In 3x = 15, the 3 does not become -3 on the right. The operation flips (\times becomes \div), not the sign. Mixing up the two rules is the most common mistake.

• "I can transpose part of a bracket." No. In 3(x + 7) = 22, you cannot transpose the +7 alone. You must either expand the bracket first (3x + 21 = 22) or transpose the whole bracket-multiplier (x + 7 = 22/3) using multiplicative transposition. Only complete factors and complete added/subtracted terms can be transposed.

• "What if c = 0 in the multiplicative case?" Then there is no transposition to do, because A \cdot 0 = 0 collapses the left side to 0, and you cannot divide by 0. The condition c \neq 0 is built into multiplicative transposition.

References

1. NCERT Class 8 Mathematics, Chapter 2: Linear Equations in One Variable — the official source of the transposition rule for Indian school students.
2. Wikipedia, Equation solving — the inverse-operation principle behind both transposition rules.
3. Wikipedia, Multiplicative inverse — why 1/c undoes multiplication by c.
4. Wikipedia, Additive inverse — why -c undoes addition of c.
5. Khan Academy, Why we do the same thing to both sides — the underlying truth that both transposition rules compress.