We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.

## Solving Equations Using the Subtraction and Addition Properties of Equality

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We began our work solving equations in previous tutorials. It has been a while since we have seen an **equation**, so we will review some of the key concepts before we go any further.

We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the **variable** that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

### Definition: Solution of an Equation

A **solution of an equation** is a value of a variable that makes a true statement when substituted into the equation.

In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

### How to Determine whether a number is a solution to an equation.

- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.

## Example

Determine whether \(y=\frac{3}{4}\) is a solution for \(4y+3=8y.\)

### Solution

Multiply. | |

Add. |

Since \(y=\frac{3}{4}\) results in a true equation, \(\frac{3}{4}\) is a solution to the equation \(4y+3=8y.\)

We introduced the Subtraction and Addition Properties of Equality earlier. We modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.

### Definition: Subtraction and Addition Properties of Equality

**Subtraction Property of Equality**

For all real numbers \(a,b,\) and \(c,\) if \(a=b,\) then \(a-c=b-c.\)

**Addition Property of Equality**

For all real numbers \(a,b,\) and \(c,\) if \(a=b,\) then \(a+c=b+c.\)

When you add or subtract the same quantity from both sides of an equation, you still have equality.

We introduced the **Subtraction Property of Equality** earlier by modeling equations with envelopes and counters. The figure below models the equation \(x+3=8.\)

The goal is to isolate the variable on one side of the equation. So we ‘took away’ \(3\) from both sides of the equation and found the **solution** \(x=5.\)

Some people picture a balance scale, as in the figure below, when they solve equations.

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

## Example

Solve: \(x+11=-3.\)

### Solution

To isolate \(x,\) we undo the addition of \(11\) by using the Subtraction Property of Equality.

Subtract 11 from each side to “undo” the addition. | ||

Simplify. | ||

Check: | ||

Substitute \(x=-14\). | ||

Since \(x=-14\) makes \(x+11=-3\) a true statement, we know that it is a solution to the equation.

In the original equation in the previous example, \(11\) was added to the \(x\), so we subtracted \(11\) to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the **Addition Property of Equality**.

## Example

Solve: \(m-4=-5.\)

### Solution

Add 4 to each side to “undo” the subtraction. | ||

Simplify. | ||

Check: | ||

Substitute \(m=-1\). | ||

The solution to \(m-4=-5\) is \(m=-1\). |

Now let’s review solving equations with fractions.

## Example

Solve: \(n-\frac{3}{8}=\frac{1}{2}.\)

### Solution

Use the Addition Property of Equality. | ||

Find the LCD to add the fractions on the right. | ||

Simplify | ||

Check: | ||

Subtract. | ||

Simplify. | ||

The solution checks. |

In Mathematics 105, we solved equations that contained decimals. We’ll review this next.

## Example

Solve \(a-3.7=4.3.\)

### Solution

Use the Addition Property of Equality. | ||

Add. | ||

Check: | ||

Substitute \(a=8\). | ||

Simplify. | ||

The solution checks. |