**3.5 Solve Equations Using Integers; The Division Property of Equality**

The topics covered in this section are:

- Determine whether an integer is a solution of an equation
- Solve equations with integers using the Addition and Subtraction Properties of Equality
- Model the Division Property of Equality
- Solve equations using the Division Property of Equality
- Translate to an equation and solve

**3.5.1 Determine Whether a Number is a Solution of an Equation**

In Solving Equations Using the Subtraction And Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

**HOW TO: 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 1**

Determine whether each of the following is a solution of $2x-5=-13$:

- $x=4$
- $x=-4$
- $x=-9$

**Solution**

Part 1. Substitute $4$ for $x$ in the equation to determine if it is true. | $2x-5=-13$ |

Substitute $4$ for $x$. | $2(4)-5 \stackrel{?}{=} -13$ |

Multiply. | $8-5 \stackrel{?}{=} -13$ |

Subtract. | $3 \neq -13$ |

Since $x=4$ does not result in a true equation, $4$ is not a solution to the equation.

Part 2. Substitute $-4$ for $x$ in the equation to determine if it is true. | $2x-5=-13$ |

Substitute $4$ for $x$. | $2(-4)-5 \stackrel{?}{=} -13$ |

Multiply. | $-8-5 \stackrel{?}{=} -13$ |

Subtract. | $-13 = -13$ |

Since $x=-4$ results in a true equation, $-4$ is a solution to the equation.

Part 3. Substitute $-9$ for $x$ in the equation to determine if it is true. | $2x-5=-13$ |

Substitute $-9$ for $x$. | $2(-9)-5 \stackrel{?}{=} -13$ |

Multiply. | $-18-5 \stackrel{?}{=} -13$ |

Subtract. | $-23 \neq -13$ |

Since $x=-9$ does not result in a true equation, $-9$ is not a solution to the equation.

**3.5.2 Solve Equations with Integers Using the Addition and Subtraction Properties of Equality**

In Solving Equations Using the Subtraction And Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

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

**PROPERTIES OF EQUALITIES**

Subtraction Property of Equality | Addition Property of Equality |
---|---|

For any number $a,b,c$, if $a=b$ then $a-c=b-c$ | For any numbers $a,b,c$, if $a=b$ then $a+c=b+c$ |

**Example 2**

Solve: $y+9=5$.

**Solution**

$y+9=5$ | |

Subtract $9$ from each side to udo the addition. | $y+9-9=5-9$ |

Simplify. | $y=-4$ |

Check the result by substituting $-4$ into the original equation.

$y+9=5$ | |

Substitute $-4$ for $y$. | $-4+9 \stackrel{?}{=} 5$ |

$5=5$✓ |

Since $y=-4$ makes $y+9=5$ a true statement, we found the solution to this equation.

**Example 3**

Solve: $a-6=-8$

**Solution**

$a-6=-8$ | |

Add $6$ to each side to undo the subtraction. | $a-6+6=-8+6$ |

Simplify. | $a=-2$ |

Check the result by substituting $-2$ into the original equation: | $a-6=-8$ |

Substitute $-2$ for $a$ | $-2-6 \stackrel{?}{=} -8$ |

$-8=-8$✓ |

The solution to $a-6=-8$ is $-2$.

Since $a=-2$ makes $a-6=-8$ a true statement, we found the solution to this equation.

**3.5.3 Model the Division Property of Equality**

All of the equations we have solved so far have been of the form $x+a=b$ or $x-a=b$. We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.

We will model an equation with envelopes and counters in the figure below.

Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

To determine the number, separate the counters on the right side into $2$ groups of the same size. So $6$ counters divided into $2$ groups means there must be $3$ counters in each group (since $6 \div 2=3$)

What equation models the situation show in the figure below? There are two envelopes, and each contains $x$ counters. Together, the two envelopes must contain a total of $6$ counters. So the equation that models the situation is $2x-6$.

We can divide both sides of the equation by $2$ as we did with the envelopes and counters.

We found that each envelope contains $3$ counters. Does this check? We know $2 \cdot 3 = 6$, so it works. Three counters in each of two envelopes does equal six.

The figure below shows another example.

Now we have $3$ identical envelopes and $12$ counters. How many counters are in each envelope? We have to separate the $12$ counters into $3$ groups. Since $12 \div 3 = 4$, there must be $4$ counters in each envelope. See the figure below.

The equation that models the situation is $3x=12$. We can divide both sides of the equation by $3$.

Does this check? It does because $3 \cdot 4 = 12$.

**Example 4**

Write an equation modeled by the envelopes and counters, and then solve it.

**Solution**

There are $4$ envelopes, or $4$ unknown values, on the left that match the $8$ counters on the right. Let’s call the unknown quantity in the envelopes $x$.

Write the equation. | $4x=8$ |

Divide both sides by $4$. | $\frac{4x}{4} = \frac{8}{4}$ |

Simplify. | $x=2$ |

There are $2$ counters in each envelope.

**3.5.4 Solve Equations Using the Division Property of Equality**

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

**DIVISION PROPERTY OF EQUALITY**

For any nuymbers $a,b,c$, and $c \neq 0$,

If $a=b$ then $\frac{a}{c} = \frac{b}{c}$.

**Example 5**

Solve: $7x=-49$.

**Solution**

To isolate $x$, we need to undo multiplication.

$7x=-49$ | |

Divide each side by $7$. | $\frac{7x}{7} = \frac{-49}{7}$ |

Simplify. | $x=-7$ |

Check the solution.

$7x=-49$ | |

Substitute $-7$ for $x$. | $7(-7) \stackrel{?}{=} -49$ |

$-49=-49$✓ |

Therefore, $-7$ is the solution to the equation.

**Example 6**

Solve: $-3y=63$.

**Solution**

To isolate $7$, we need to undo multiplication.

$-3y=63$ | |

Divide each side by $-3$. | $\frac{-3y}{-3} = \frac{63}{-3}$ |

Simplify. | $x=-21$ |

Check the solution.

$-3y=63$ | |

Substitute $-21$ for $y$. | $-3(-21) \stackrel{?}{=} 63$ |

$63=63$✓ |

Since this is a true statement, $y=-21$ is the solution to the equation.

**3.5.5 Translate to an Equation and Solve**

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

**Example 7**

Translate and solve: five more than $x$ is equal to $-3$.

**Solution**

five more than $x$ is equal to $-3$ | |

Translate | $x+5=-3$ |

Subtract $5$ from both sides. | $x+5-5=-3-5$ |

Simplify. | $x=-8$ |

Check the answer by substituting it into the original equation.

$x+5=-3$

$-8+5 \stackrel{?}{=} -3$

$-3=-3$✓

**Example 8**

Translate and solve: the difference of $n$ and $6$ is $-10$.

**Solution**

the difference of $n$ and $6$ is $-10$ | |

Translate. | $n-6=-10$ |

Add $6$ to each side. | $n-6+6=-10+6$ |

Simplify. | $n=-4$ |

Check the answer by substituting it into the original equation.

$n-6=-10$

$-4-6 \stackrel{?}{=} -10$

$-10=-10$✓

**Example 9**

Translate and solve: the number $108$ is the product of $-9$ and $y$.

**Solution**

the number of $108$ is the product of $-9$ and $y$ | |

Translate. | $108=-9y$ |

Add $6$ to each side. | $\frac{108}{-9} = \frac{-9y}{-9}$ |

Simplify. | $-12=y$ |

Check the answer by substituting it into the original equation.

$108=-9y$

$108 \stackrel{?}{=} -9(-12)$

$108=108$✓

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*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/3-5-solve-equations-using-integers-the-division-property-of-equality*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*