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# 4.11: Simplify Variable Expressions Involving Integer Multiplication

Difficulty Level: At Grade Created by: CK-12
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Practice Simplify Variable Expressions Involving Integer Multiplication

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Have you ever been skiing?

In Alaska, Molly and her family went on a skiing trip. There are four people in Molly's family. When they got there, they saw that four other families of four were also there. Molly isn't sure how much a lift ticket costs. This is an unknown variable in this situation.

Can you write a variable expression involving integer multiplication to figure out how many people will need lift tickets?

Well, if you aren't sure how to do this, don't worry, you will learn all about it in this Concept.

### Guidance

Do you remember what a variable expression is?

A variable expression is a math phrase using numbers, operations and variables. A variable expression can also contain like terms. A like term is a term that is common between one or more terms in the equation. When you have like terms, we can combine them using addition and subtraction.

Pay attention! Here is a change!

How does this happen?

Find the value of this expression 3z(2)\begin{align*}3z \cdot (-2)\end{align*}

The Commutative Property of Multiplication states that the order in which factors are multiplied does not matter.

3z(2)=23z\begin{align*}3z \cdot (-2) = -2 \cdot 3z\end{align*}

The Associative Property of Multiplication states that you can group the factors being multiplied in any order.

23z=23z=(23)z\begin{align*}-2 \cdot 3z = -2 \cdot 3 \cdot z = (-2 \cdot 3) \cdot z\end{align*}

Now we can multiply the integers. Since -2 and 3z\begin{align*}3z\end{align*} have different signs, the product will be negative.

23=6\begin{align*}2 \cdot 3=6\end{align*}, so (2)3=6\begin{align*}(-2)\cdot 3=-6\end{align*} and (23)z=6z\begin{align*}(-2 \cdot 3)\cdot z = -6z\end{align*}

The value of the expression is 6z\begin{align*}-6z\end{align*}.

Find the value of this expression (5)(2m)(n)\begin{align*}(-5)(-2m)(n)\end{align*}

The Associative Property of Multiplication states that you can group the factors being multiplied in any order.

(5)(2m)(n)=(5)(2)mn\begin{align*}(-5)(-2m)(n) = (-5)\cdot (-2) \cdot m \cdot n\end{align*}

Now we can multiply the integers. Since -5 and -2 are both negative, the product will be positive.

52=52=10\begin{align*}-5 \cdot -2=5 \cdot 2 = 10\end{align*}, so (5)(2)mn=10mn=10mn\begin{align*}(-5) \cdot (-2) \cdot m \cdot n = 10 \cdot mn=10mn\end{align*}

The value of the expression is 10mn\begin{align*}10mn\end{align*}.

Multiply the following variable expressions.

#### Example A

3x(4y)\begin{align*}3x(4y)\end{align*}

Solution:12xy\begin{align*}12xy\end{align*}

#### Example B

6a(4b)\begin{align*}-6a(-4b)\end{align*}

Solution:24ab\begin{align*}24ab\end{align*}

#### Example C

4z(10)\begin{align*}-4z(10)\end{align*}

Solution:40z\begin{align*}-40z\end{align*}

Here is the original problem once again.

In Alaska, Molly and her family went on a skiing trip. There are four people in Molly's family. When they got there, they saw that four other families of four were also there. Molly isn't sure how much a lift ticket costs. This is an unknown variable in this situation.

Can you write a variable expression involving integer multiplication to show how many people will need lift tickets?

To write this variable expression, let's write down what we know first.

We know that there are four people in Molly's family who will all need lift tickets.

4x\begin{align*}4x\end{align*}

This is the first part of the variable expression because we don't know the price of the lift ticket.

Then there are four groups like Molly's family. We can multiply the variable expression by 4.

4(4x)\begin{align*}4(4x)\end{align*}

Now we can simplify.

16x\begin{align*}16x\end{align*}

This is the number of people who will need lift tickets.

### Vocabulary

Here are the vocabulary words in this Concept.

Integer
the set of whole numbers and their opposites.
Product
the answer in a multiplication problem
Factors
the numbers being multiplied
Variable Expression
a number phrase using numbers, operations and variables.
Commutative Property of Multiplication
states that the order that we multiply terms does not change the product.
Associative Property of Multiplication
states that changing the grouping of factors does not change the product.

### Guided Practice

Here is one for you to try on your own.

The temperature outside Fred's house is dropping at a rate of 2F\begin{align*}2^\circ F\end{align*} each hour. Represent the total change in the temperature over the next 5 hours as an integer.

First, let's write an expressing to represent the situation.

(2)(5)=10\begin{align*}(-2)(5) = -10\end{align*}

The temperature change was 10\begin{align*}-10^\circ\end{align*}.

### Video Review

Here is a video for review.

### Practice

Directions: Multiply each variable expression.

1. (7k)(6)\begin{align*}(-7k)(-6)\end{align*}

2. (8)(3a)(b)\begin{align*}(-8)(3a)(b)\end{align*}

3. 6a(b)(c)\begin{align*}-6a(b)(c)\end{align*}

4. 8a(6b)\begin{align*}-8a(6b)\end{align*}

5. (12y)(3x)(1)\begin{align*}(12y)(-3x)(-1)\end{align*}

6. 8x(4)\begin{align*}-8x(4)\end{align*}

7. a(5)(4b)\begin{align*}-a(5)(-4b)\end{align*}

8. 2ab(12c)\begin{align*}-2ab(12c)\end{align*}

9. 12ab(12c)\begin{align*}-12ab(12c)\end{align*}

10. 8x(12z)\begin{align*}8x(12z)\end{align*}

11. 2a(14c)\begin{align*}-2a(-14c)\end{align*}

12. 12ab(11c)\begin{align*}-12ab(11c)\end{align*}

13. 22ab(2c)\begin{align*}-22ab(-2c)\end{align*}

14. 18ab(12)\begin{align*}18ab(12)\end{align*}

15. 21a(3b)\begin{align*}-21a(-3b)\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Associative property

The associative property states that the order in which three or more values are grouped for multiplication or addition will not affect the product or sum. For example: $(a+b) + c = a + (b+c) \text{ and\,} (ab)c = a(bc)$.

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.

Factors

Factors are numbers or values multiplied to equal a product.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Product

The product is the result after two amounts have been multiplied.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

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