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How can an equation be solved while keeping both sides equal?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

To solve an algebraic equation while maintaining equality, you must apply the same operations to both sides of the equation. The goal is to isolate the variable by using inverse operations, which effectively undo the effects of the operations currently applied to the variable.12

Because an equation represents a statement of equality between two expressions, any change made to one side must be mirrored on the other to preserve that balance. For example, if you add, subtract, multiply, or divide by a specific value on one side, you must perform the exact same action on the other side.3

By systematically applying these inverse procedures—such as using subtraction to undo addition or division to undo multiplication—you can simplify the equation until the variable is isolated, revealing the root or solution that satisfies the original statement.312