# Equations in Mathematics

## Definition

An **equation** is a statement that expresses the equality of two mathematical expressions. An equation has an equal sign, a right side expression and a left side expression.

Examples of equations

3x + 3 = 2x + 4 : the left side of the equation is the expression 3x + 3 and the right side is 2x + 4.

2x + 3y = 2 - 2x : equation in two variables x and y.

## Solutions of an Equation

If we substitute x by -3 in the equation 2x + 8 = -2x - 4, we obtain

left side: 2x + 8 = 2(-3) + 8 = -6 + 8 = 2

right side: -2x - 4 = -2(-3) - 4 = 6 - 4 = 2

Since a substitution of x = - 3 in the equation gives a true statement 2 = 2, we call -3 the **solution** or **root** of the given equation 2x + 8 = -2x - 4. The set of all solutions of an equation is called the **solution set** of the equation.

To solve an equation is to find all its solutions.

## Equivalent Equations

Equations are **equivalent** if they have exactly the same solutions.

The following equations are equivalent since they have the same solution x = 0.

-3x + 2 = x + 2

-3x = x

x = 0
## Properties of Equality

### 1 - Addition Property of Equality

If we add the same number (or mathematical expression) to both sides of an equation, we do not change the solution set of the equation.

**If A = B then A + C = B + C**

Example

The equation **2x + 3 = 5**

and the equation **2x + 3 + (-3) = 5 + (-3)** have the same solution x = 1.

### 2 - Multiplication Property of Equality

If we multiply both sides of an equation by the same number (or mathematical expression), we do not change the solution set of the equation.

**If A = B then C × A = C × B** , with C not equal to zero.

Example

The equation **x / 2 = 4**

and the equation **2 * (x / 2) = 2 * 4** have the same solution x = 8.

## More References and links

Solve Equations, Systems of Equations and Inequalities.