# Solving an equation using addition and subtraction

There are four guiding principles which will help you in solving equations.

### Principle 1: A number combined with its opposite in sign is equal to zero.

#### For example:

[math]2 - 2 = 0[/math] | [math]-2 + 2 = 0[/math] | [math]0 = -10 + 10[/math] | [math]0 = 64 - 64[/math] |

### Principle 2: Zero added to any number is equal to that number.

#### For example:

[math]6 + 0 = 6 [/math] | [math]0 + 12 = 12[/math] | [math]x + 0 = x[/math] | [math]0 + y = y[/math] |

### Principle 3: You can only combine like terms:

- numbers are added to numbers, and
- variables are added to variables.

#### For example:

[math]5 + x + 7 = 12 + x \quad \text{or} \quad x + 12[/math] | [math] x + 5 - 5 = x + 0 = x[/math] | [math]6 + x - 6 = 0 + x = x[/math] |

## Solving an equation

A **variable** is the unknown in an equation represented by a letter.

Solving an equation is like unwrapping a package, we unwrap the variable. In other words, we undo what was done to the variable. So, if 6 were added to the variable, we subtract 6 to get to the variable and find out what number it represents. Similarly, if 2 were subtracted from the variable, we would add 2 to “unwrap” the variable. The thing is, you must remember that if you perform an operation to one side of an equation, you must do the same thing to the other side.

#### Example 1: [math]x + 5 = 85[/math]

Look on the side of the equation containing the variable. Since 5 was added to the variable x, we subtract 5 from both sides to get the x alone on one side of the equal sign. Then we can say that x = such and such.

[math]\begin{align} x + 5 &= 85 \\ x + 5 - 5 &= 85 - 5 \qquad \text{subtract 5 from both sides} \\ x + 0 &= 80 \qquad \text{combine -5 and 5, and 85 and -5}\\ x &= 80 \\ \end{align}[/math]

#### Example 2: [math]x -4 = 12[/math]

The variable is on the left side of the equation. Since 4 is subtracted from the variable, we will undo that by adding 4 to both sides. Then we can say that x = 16.

[math]\begin{align} x - 4 &= 12 \\ x - 4 + 4 &= 12 + 4 \qquad \text{add 4 to both sides} \\ x + 0 &= 16 \qquad \text{typically, we skip this step} \\ x &= 16 \\ \end{align}[/math]

### Example 3: [math]24 = x + 14[/math]

In this example the variable is on the right side of the equation.

Since 14 is added to the variable, we will subtract 14 from both sides of the equation.

[math]\begin{align} 24 & = x + 14 \\ 24 -14 & = x + 14 - 14 \\ 10 & = x \qquad \text{or} \qquad x = 10 \\ \end{align}[/math]

So, basically you are combining positives and negatives of a particular number to get them to cancel out to get the variable alone on one side of the equal sign.

### Example 4: [math]-15 + x = 34[/math]

The variable is on the left side of the equation with a -15 so you will add 15 to both sides.

[math]\begin{align} -15 + x & = 34 \\ -15 + 15 + x & = 34 + 15 \\ x & = 49 \\ \end{align}[/math]

### Example 5: [math]55 = -10 + x[/math]

The variable is on the right side of the equation, along with a -10.

[math]\begin{align} 55 & = -10 + x \\ 55 + 10 & = -10 + 10 + x \\ 65 & = x \qquad \text{or} \qquad x = 65\\ \end{align}[/math]