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Equivalent Equations & Inverse Operations

The engine of solving is one balance rule: do the same thing to both sides and you keep the solution. Pair that with inverse operations and you can isolate the variable, step by step, on purpose.

Keep the balance, keep the solution

Picture an equation as a balanced scale: the equals sign says both pans weigh the same. If you add the same weight to both pans, or remove the same weight from both, the scale stays level. In algebra this means you may add, subtract, multiply, or divide both sides by the same thing (never dividing by zero) and the equation still has exactly the same solutions. Two equations with the same solutions are called equivalent equations.

Inverse operations undo what was done

To free the variable, you peel off whatever surrounds it using an inverse operation — the operation that undoes it. Addition and subtraction undo each other; multiplication and division undo each other. Subtracting a number uses its additive inverse (its negative); dividing by a number uses its multiplicative inverse (its reciprocal).

Order matters. To get x by itself, undo the operations in reverse of the order of operations: clear the additions and subtractions first, then the multiplications and divisions last. That is the opposite order from how you would evaluate the expression.

Solve   3x + 5 = 20

Undo the +5 first (subtract 5 from both sides):
  3x + 5 - 5 = 20 - 5
  3x = 15
Undo the times-3 (divide both sides by 3):
  3x / 3 = 15 / 3
  x = 5

Check in the original: 3(5) + 5 = 15 + 5 = 20  TRUE.
Subtraction undoes the +5; division undoes the times-3 — additions before multiplications.

Transposition is a shortcut, not a new rule

You will often see a term “moved” from one side to the other with its sign flipped — for example, x + 7 = 12 becoming x = 12 - 7. This is transposition, and it is nothing mysterious: moving +7 across and changing it to -7 is exactly the same as subtracting 7 from both sides. The shortcut is fine to use once you trust the balance rule behind it.