Solving One-Step Equations

Inverse operations are two operations that, all else being equal, undo one another. For instance, adding $6$ and subtracting $6$ are inverse operations because they cancel each other out. $\begin{aligned} &x+6\\ &x+6{\color{#0000FF}{\ -\ 6}}\\ &x \end{aligned}$ In an expression like $x+6,$ the addition of $6$ to $x$ is eliminated by performing the inverse operation: a subtraction of $6.$ Using inverse operations on an equation, however, is a little different: in order to adhere to the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side to maintain equality. $\begin{aligned} x\div{2}&=1\\ x\div{2}{\color{#0000FF}{\ \times{\ 2}}}&=1{\color{#0000FF}{\ \times{\ 2}}}\\ x&=2 \end{aligned}$

The Addition Property of Equality states that adding the same number to each side of an equation yields an equivalent equation.

The Subtraction Property of Equality is similar to the Addition one, with the exception being the sign used — subtracting the same number from each side of an equation yields an equivalent equation.

Either of these properties can be used when eliminating a term in an equation. Deciding which to use depends on if the term is being added to or subtracted from the variable.

The Multiplication Property of Equality states that multiplying each side of an equation by the same number yields an equivalent equation.

Similarly, by the Division Property of Equality, an equivalent equation is produced when each side of an equation is divided by the same non-zero number.

These two properties can be used when solving equations to eliminate denominators and factors, since multiplication is the inverse operation of division and vice versa.