# 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:

 $2 - 2 = 0$ $-2 + 2 = 0$ $0 = -10 + 10$ $0 = 64 - 64$

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

#### For example:

 $6 + 0 = 6$ $0 + 12 = 12$ $x + 0 = x$ $0 + y = y$

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

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

#### For example:

 $5 + x + 7 = 12 + x \quad \text{or} \quad x + 12$ $x + 5 - 5 = x + 0 = x$ $6 + x - 6 = 0 + x = x$

## 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:   $x + 5 = 85$

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.

\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}

#### Example 2:   $x -4 = 12$

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.

\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}

### Example 3:   $24 = x + 14$

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.

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

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:   $-15 + x = 34$

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

\begin{align} -15 + x & = 34 \\ -15 + 15 + x & = 34 + 15 \\ x & = 49 \\ \end{align}

### Example 5:   $55 = -10 + x$

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

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