 # 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.