Foundations of Algebra

Overview

An equation says that two expressions represent the same value. Solving it means finding the value of the unknown that makes the statement true. The central rule is to preserve equality: if you add, subtract, multiply or divide on one side, apply the same valid operation to the other side.

Start by choosing a symbol and defining its meaning with a unit. If x is the number of items, a charge of a fixed fee plus a cost per item can be written as a constant plus a multiple of x. Translate the full relationship before simplifying it. This prevents a plausible-looking equation from silently changing the original problem.

When checking a solution, substitute the proposed value into the equation before any rearrangement. A solution that satisfies an intermediate line but fails the original statement indicates an invalid transformation or arithmetic error. Division by zero is never permitted; some equations have no solution, and others hold for every value rather than for one specific value.

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• Linear Equations in One Variable

  Use equal operations on both sides of an equation, then substitute the result into the original relationship.