Solving Simple Equations

When solving a simple equation, think of the equation as a balance, with the equals sign (=) being the fulcrum or center. Thus, if you do something to one side of the equation, you must do the same thing to the other side. Doing the same thing to both sides of the equation (say, adding 3 to each side) keeps the equation balanced.

Solving an equation is the process of getting what you're looking for, or solving for, on one side of the equals sign and everything else on the other side. You're really sorting information. If you're solving for x, you must getx on one side by itself.

Addition and subtraction equations

Some equations involve only addition and/or subtraction.

Example 1

Solve for x.

x + 8 = 12

To solve the equation x + 8 = 12, you must get x by itself on one side. Therefore, subtract 8 from both sides.

equation

To check your answer, simply plug your answer into the equation:

 equation

Example 2

Solve for y.

y – 9 = 25

To solve this equation, you must get y by itself on one side. Therefore, add 9 to both sides.

equation

To check, simply replace y with 34:

 equation

Example 3

Solve for x.

x + 15 = 6

To solve, subtract 15 from both sides.

equation

To check, simply replace x with –9 :

 equation

Notice that in each case above, opposite operations are used; that is, if the equation has addition, you subtract from each side.

Multiplication and division equations

Some equations involve only multiplication or division. This is typically when the variable is already on one side of the equation, but there is either more than one of the variable, such as 2 x, or a fraction of the variable, such as

equation or equation

In the same manner as when you add or subtract, you can multiply or divide both sides of an equation by the same number, as long as it is not zero, and the equation will not change.

Example 4

Solve for x.

x = 9

Divide each side of the equation by 3.

equation

To check, replace x with 3:

 equation

Example 5

Solve for y.

equation

To solve, multiply each side by 5.

equation

To check, replace y with 35:

 equation

Example 6

Solve for x.

equation

To solve, multiply each side by equation.

equation

Or, without canceling,

 equation

Notice that on the left you would normally not write equation because it would always cancel to 1 x, or x.

equation

Combinations of operations

Sometimes you have to use more than one step to solve the equation. In most cases, do the addition or subtraction step first. Then, after you've sorted the variables to one side and the numbers to the other, multiply or divide to get only one of the variables (that is, a variable with no number, or 1, in front of it: x, not 2 x).

Example 7

Solve for x.

x + 4 = 10

Subtract 4 from both sides to get 2 x by itself on one side.

equation

Then divide both sides by 2 to get x.

equation

To check, substitute your answer into the original equation:

 equation

Example 8

Solve for x.

5x – 11 = 29

Add 11 to both sides.

equation

Divide each side by 5.

equation

To check, replace x with 8:

 equation

Example 9

Solve for x.

equation

Subtract 6 from each side.

equation

Multiply each side by equation.

equation

To check, replace x with 9: 

equation

Example 10

Solve for y.

equation

Add 8 to both sides.

equation

Multiply each side by equation.

equation

To check, replace y with –25: 

equation

Example 11

Solve for x.

x + 2 = x + 4

Subtract 2 from both sides (which is the same as adding –2).

equation

Subtract x from both sides.

equation

Note that 3 x – x is the same as 3 x – 1 x.

Divide both sides by 2.

equation

To check, replace x with 1:

 equation

Example 12

Solve for y.

y + 3 = 2 y + 9

Subtract 3 from both sides.

equation

Subtract 2 y from both sides.

equation

Divide both sides by 3.

equation

To check, replace y with 2:

 equation

Sometimes you need to simplify each side (combine like terms) before actually starting the sorting process.

Example 13

 Solve for x.     

x + 4 + 2 = 12 + 3

First, simplify each side.

equation

Subtract 6 from both sides.

equation

Divide both sides by 3.

equation

To check, replace x with 3: 

equation

Example 14

Solve for x.

x + 2 x + 4 = 5 x + 3 + 11

Simplify each side.

x + 4 = 5 x + 14

Subtract 4 from both sides.

equation

Subtract 5 x from both sides.

equation

To check, replace x with 10: 

equation