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4.11: 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 know 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 \cdot (-2)

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

3z \cdot (-2) = -2 \cdot 3z

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

-2 \cdot 3z = -2 \cdot 3 \cdot z = (-2 \cdot 3) \cdot z

Now we can multiply the integers. Since -2 and 3z have different signs, the product will be negative.

2 \cdot 3=6 , so (-2)\cdot 3=-6 and (-2 \cdot 3)\cdot z = -6z

The value of the expression is -6z .

Find the value of this expression (-5)(-2m)(n)

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

(-5)(-2m)(n) = (-5)\cdot (-2) \cdot m \cdot n

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

-5 \cdot -2=5 \cdot 2 = 10 , so (-5) \cdot (-2) \cdot m \cdot n = 10 \cdot mn=10mn

The value of the expression is 10mn .

Multiply the following variable expressions.

Example A

3x(4y)

Solution: 12xy

Example B

-6a(-4b)

Solution: 24ab

Example C

-4z(10)

Solution: -40z

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

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)

Now we can simplify.

16x

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

Vocabulary

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.
Factors
Factors are the numbers being multiplied to equal a product.
Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.
Commutative Property of Multiplication
A property that states that it doesn’t matter which order you multiply terms. The product will be the same. ab = ba .
Associative Property of Multiplication
The associative property of multiplication states that changing the grouping of numbers does not affect the product of the numbers.

Guided Practice

Here is one for you to try on your own.

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

Answer

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

(-2)(5) = -10

The temperature change was -10^\circ .

Video Review

This is a James Sousa video on multiplying integers. It is a supporting video for this Concept.

Explore More

Directions: Multiply each variable expression.

1. (-7k)(-6)

2. (-8)(3a)(b)

3. -6a(b)(c)

4. -8a(6b)

5. (12y)(-3x)(-1)

6. -8x(4)

7. -a(5)(-4b)

8. -2ab(12c)

9. -12ab(12c)

10. 8x(12z)

11. -2a(-14c)

12. -12ab(11c)

13. -22ab(-2c)

14. 18ab(12)

15. -21a(-3b)

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Date Created:

Nov 30, 2012

Last Modified:

Nov 26, 2014
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