Linear Equations and Equivalent Formulas

Foundations of Linear Equations and Inverse Operations

An equation states that two mathematical expressions are equal in value. A linear equation in one variable can be simplified to ax + b = cx + d, where a, b, c, and d are constants; having no powers above one and no variable in a denominator is not, by itself, enough to make an equation linear. When an equation has one solution, solving it means isolating the variable to find its value. To isolate that variable, you must maintain balance across the equals sign. To keep an equation equivalent, add or subtract the same value on both sides, or multiply or divide both sides by the same nonzero value.

The primary tools for isolating a variable are inverse operations, which are mathematical operations that undo each other. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. Therefore, to solve an equation where a number is added to a variable, subtract that number from both sides. To solve an equation where a number is subtracted from a variable, add that number to both sides. Similarly, to isolate a variable multiplied by a nonzero coefficient, divide both sides by that coefficient, and to isolate a variable divided by a nonzero number, multiply both sides by that number.

Careful attention to positive and negative signs is critical when performing these inverse operations. Dividing or multiplying two numbers with the same sign yields a positive result, whereas dividing or multiplying two numbers with different signs yields a negative result. In some cases, simplifying an algebraic statement involves addressing signs across the entire relationship: multiplying or dividing an entire equation by negative one reverses the sign of every term.