An equation is a mathematical statement that asserts the equivalence of two expressions. In modern notation, this is indicated by placing the expressions on either side of an equal sign (=). For example, $x+3=5$ asserts that $x+3$ is equal to 5.

Equations often express relationships between given quantities ("knowns") and quantities yet to be determined ("unknowns"). By mathematical convention, unknowns are denoted by letters toward the end of the alphabet $(x,y,z...)$, while knowns are denoted by letters at the beginning of the alphabet $(a,b,c...)$. 

The process of expressing an equation's unknowns in terms of its knowns is called solving the equation. In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation. 

If an equation in algebra is known to be true, the following properties may be used to produce another true equation. For each property, both the formal definition and the plain-English definition are provided.

The Addition Property of Equality

If $a=b$, then $a+c=b+c$.

In other words, any real number can be added to both sides of an equation.

The Subtraction Property of Equality

If $a=b$, then $a-c=b-c$.

In other words, any real number can be subtracted from both sides of an equation.

The Multiplication Property of Equality

If $a=b$, then $ca=cb$.

In other words, any real number can be multiplied to both sides of an equation.

The Division Property of Equality

If $a=b$, and $c\neq 0$, then $\dfrac{a}{c}=\dfrac{b}{c}$.

In other words, any non-zero real number can divide both sides of an equation.

Solving Equations using the Properties of Equality

Example 1

The bill for the repair of a car was $458, and the cost of parts was $339.  The cost of labor was $34 per hour. Write and solve an equation to find the number of hours of labor that went into the repair.

Let $x$ equal the unknown value: the number of hours of labor. The equation is therefore:

$34x+339=458$.  

In English, the cost of the labor ($34) multiplied by the number of hours of labor $(x)$, plus the cost of the parts ($339), is equal to the total bill for the repair ($458).

To solve for the unknown, first undo the addition operation (using the subtraction property) by subtracting $339 from both sides of the equation: 

$\begin{aligned} 34x+339-339&=458-339 \\ 34x&=458-339 \\ 34x&=119 \end{aligned}$

Then undo the multiplication operation (using the division property) by dividing both sides of the equation by 34: 

$\begin{aligned} \dfrac{34x}{34}& =\dfrac{119}{34} \\ x& =\dfrac{119}{34} \\ x&=3.5 \end{aligned}$

This means the car repair labor took 3.5 hours.

Example 2

Solve the following equation using properties of equality:

$\dfrac{1}{8}x-5=3$

First, use the addition property to add 5 to both sides of the equation:

$\begin{aligned} \dfrac{1}{8}x-5+5&=3+5 \\ \dfrac{1}{8}x&=3+5 \\ \dfrac{1}{8}x&=8 \end{aligned}$

Second, use the multiplication property to multiply both sides of the equation by 8:

$\begin{aligned} 8 \cdot \dfrac{1}{8}x&= 8 \cdot 8 \\ x&=64 \end{aligned} $

This is the solution to the equation.