Cross multiply

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In an equation in which two fractions or rational expressions are set equal, we can cross multiply provided neither denominator is zero. That is, if b and d are non-zero, then

(equation one)

if and only if

ad = bc (equation two).

To prove this, we use the rule that both sides of an equation can be multiplied by any non-zero number. If b and d are both non-zero, then so is bc. Multiplying both sides of equation one by bd yields

.

Reducing to lowest terms gives equation two.

Conversely, starting with

ad = bc

we can divide both sides of the equation by bd yielding

and reducing to lowest terms gives equation one.

Often this step is the first step in discovering whether or not two fractions are equal, or in solving an equation containing rational expressions.

More generally, any equation containing fractions or rational expressions can be simplified by multiplying both sides by the least common denominator. This step is called "clearing fractions".