Mathématiques

Why Do You Do the Same Thing to Both Sides of an Equation?

An equals sign claims both sides are the same number. Change one side only and the claim stops being true. The safe moves, and three that cost the answer.

Réponse courte

Why do you do the same thing to both sides of an equation?

Because an equals sign claims that the two sides are the same number. Doing something to one side only breaks that claim, so the next line of working is about a different problem.

La réponse en bref
An equation says the two sides are equal - the same number written two ways. Any operation applied to both sides keeps them equal, so the new equation has the same solution as the old one. Apply it to one side only and the equality is destroyed, along with the answer you were looking for.

Ce qui change la réponse

Multiplying both sides by zero is perfectly legal and ruins the equation, because 0 = 0 is true for every value and tells you nothing.
Squaring both sides preserves the equation but can add solutions that do not satisfy the original, so every answer has to be checked.
Dividing both sides by an expression containing the unknown can lose a solution, because you may be dividing by zero without knowing it.

What an equals sign is actually claiming

3x + 5 = 20 is a sentence, and like any sentence it is either true or false. It claims that for the number x stands for, three of it plus five comes to twenty. Solving is the work of finding the value that makes the sentence true, and everything you do to the equation has to keep it saying the same thing about that value.

That is the entire justification for the rule. Subtracting 5 from both sides gives 3x = 15, a different sentence with the same solution, x = 5. Subtracting 5 from the left only gives 3x = 20, which is a sentence about a completely different number - and that number, 20/3, is the answer to nothing anyone asked.

The scales picture is a good one as long as it is used properly. Two pans balance because their contents weigh the same. Take five grams off one pan and it tips, and from that moment the pans tell you nothing about each other. The equals sign is the balance point, not a full stop.

The move that produces the commonest wrong answer

The error is rarely as blatant as changing one side and leaving the other. It is subtler than that: the operation is applied to both sides, but to only part of one of them.

Take x/2 + 3 = 5. Multiplying both sides by 2 should give x + 6 = 10, because everything on the left gets multiplied, the 3 included. What gets written instead is x + 3 = 10, giving x = 7 where the answer is x = 4. Both sides were multiplied. One of them was only half multiplied.

Brackets make the same point from the other direction. From 2(x + 4) = 18, dividing both sides by 2 gives x + 4 = 9, which is right, because the 2 was multiplying the whole bracket. From 2x + 4 = 18, dividing both sides by 2 must give x + 2 = 9, not x + 4 = 9, because the 4 is a separate term and it has to be halved too. The difference is what the operation is being applied to: a single product in one case, a sum in the other.

Substitution catches this every time and takes ten seconds. Put your answer back into the original equation - not into a later line, which may already carry the error - and check that the two sides come to the same number. x = 7 in x/2 + 3 gives 6.5, not 5, and the error is found.

Three operations that are legal and still cost you the answer

Doing the same thing to both sides is necessary but not sufficient. Three operations obey the rule perfectly and still damage the equation, and all three appear on Higher and Extended papers.

The pattern behind all three is worth naming. Adding, subtracting, and multiplying or dividing by a non-zero number are reversible - you can undo them and get back exactly where you were - and that is precisely why they preserve the set of solutions. Squaring and multiplying by zero cannot be undone, and every trap in the list is a version of that one fact.

Multiplying both sides by zero. From 3x + 5 = 20 you get 0 = 0, which is true for every value of x. Nothing was broken; the information was simply thrown away.
Squaring both sides. From x = 3 you get x² = 9, whose solutions are 3 and -3. The squaring added a solution, which is why answers to equations containing square roots must be substituted back and some of them rejected.
Dividing both sides by an expression containing the unknown. From x² = 5x, dividing by x gives x = 5 and silently loses x = 0, which was also a solution. Factorise instead: x² - 5x = 0, so x(x - 5) = 0, and both solutions survive.

The same principle with one extra rule

Inequalities are handled the same way and for the same reason. A statement that one side is smaller than the other stays true if both sides are shifted by the same amount, or scaled by the same positive amount.

The single exception is multiplying or dividing by a negative number, which reverses the order of the two sides and therefore requires the sign to be turned round. Everything else transfers unchanged - add to both sides, subtract from both sides, multiply both sides by a positive number, and the statement holds.

Questions fréquentes

Why not just move a number to the other side and change its sign?

You can, and it gives the right answer - because it is a description of what doing the same to both sides produces, not a separate rule. The risk is that a student who has learned only the shortcut has nothing to fall back on when the unknown appears on both sides, or when the operation is a division rather than an addition.

What do you do when x appears on both sides?

The same thing. Subtract the smaller x term from both sides so that the unknown ends up on one side only. From 5x + 2 = 3x + 10, subtracting 3x from both sides gives 2x + 2 = 10, then subtracting 2 gives 2x = 8 and x = 4. Choosing the smaller term keeps the coefficient positive.

How do you check an answer to an equation?

Substitute it into the original equation and work out each side separately. If they come to the same number, the answer is right. Checking against a later line of your own working proves nothing, because any error you made is already in it.

Does the balance scale picture work for every equation?

Only as far as adding and multiplying by positive numbers. It gives no picture at all of squaring both sides, of multiplying by a negative, or of dividing by an unknown - which is where the genuine traps are. It is a good first model and it needs replacing with the reversibility idea by the time quadratics arrive.

Should you expand brackets before doing anything to both sides?

You do not have to; both orders work as long as the operation is applied to the whole side. 2(x + 4) = 18 can be divided by 2 first, giving x + 4 = 9, or expanded first to 2x + 8 = 18. Both give x = 5. Dividing first is usually quicker when the bracket has a factor that divides cleanly.

Dernière mise à jour 21 août 2026