**Solving Simple Equations**

An **equation** is a statement of equality of two algebraic expressions: e.g., 3 + 2 = 5 is an equation. An equation can contain one or more variables: e.g., x + 7 = 10 is an equation with one variable, x. If an equation has no variables, it is called a **closed** equation. Closed equations can be either true or false: e.g., 6 + 8 = 14 is a closed true equation, whereas 3 + 5 = 6 is a false closed equation. Open equations, also called **conditional** equations, are neither true nor false. However, if a value for the variable is substituted in the equation and a closed true equation results, the value is called a **solution** or **root** of the equation. For example, when 3 is substituted for x in the equation x + 7 = 10, the resulting equation 3 + 7 = 10 is true, so 3 is called a solution of the equation. Finding the solution of an equation is called **solving** the equation. The expression to the left of the equal sign in an equation is called the **left member** or **left side** of the equation. The expression to the right of the equal sign is called the **right member** or **right side** of the equation.

**Four Principles for Solving Simple Equations**

In order to solve an equation, it is necessary to transform the equation into a simpler equivalent equation with only the variable on one side and a constant on the other side. There are four basic types of equations and four principles that are used to solve them. These principles do not change the nature of an equation: i.e., the simpler equivalent equation has the same solution as the original equation.

In order to **check** an equation, substitute the value of the solution or root for the variable in the original equation and see if a closed true equation results.

**The Addition Principle**

An equation such as x – 8 = 22 can be solved by using the **addition principle** . *The same number can be added to both sides of an equation without changing the nature of the equation* .

**Example 1**

Solve x – 8 = 22.

**Solution 1**

Check:

**The Subtraction Principle**

An equation such as x + 5 = 9 can be solved by the **subtraction principle** . The same number can be subtracted from both members of the equation without changing the nature of the equation.

**Example 2**

Solve x + 5 = 9.

**Solution 2**

Check:

**The Division Principle**

An equation such as 7x = 28 can be solved by using the **division principle.** *Both sides of an equation can be divided by the same non-zero number without changing the nature of the equation* .

**Example 3**

Solve 7x = 28.

**Solution 3**

Check:

**The Multiplication Principle**

An equation such as can be solved by using the **multiplication principle.** *Both sides of an equation can be multiplied by the same non-zero number without changing the nature of the equation* .

**Example 4**

Solve

**Solution 4**

Check:

As you can see, there are four basic types of equations and four basic principles that are used to solve them. Before attempting to solve an equation, you should see what operation is being performed on the variable and then use the opposite principle to solve the equation. Addition and subtraction are opposite operations and multiplication and division are opposite operations.

Find practice problems and solutions at Solving Equations Practice Problems - Set 1.

**Solving Equations Using Two Principles**

Most equations require you to use more than one principle to solve them. These equations use the addition or subtraction principle first and then use the division principle.

**Examples**

**Example 1**

Solve 8x + 5 = 37.

**Solution 1**

Check:

**Example 2**

Solve 4x + 20 = –4.

**Solution 2**

Check:

**Example 3**

Solve –3x + 12 = –36.

**Solution 3**

Check:

Find practice problems and solutions at Solving Equations Practice Problems - Set 2.

More practice problems for this concept can be found at: Expressions And Equations Practice Test.

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