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Lesson: Linear Models and the Distributive Property

Write an Expression Using the Distributive Property over Multiplication

Some expressions representing real-world situations can be easily written in the form [[ a bx+c ]]. When solving problems with expressions written in this form, you will use the distributive property over multiplication.

Example:

You and your friend are selling raffle tickets for the local rotary club. Tickets are sold for $2 each. Your friend has already sold 100 tickets, and you are going to sell the remaining tickets.

Write an expression for the total amount made from raffle ticket sales.

In this problem, you are comparing two quantities: the number of tickets that you sell and the total amount of money made from ticket sales. In the first row of the table, write the column headings to describe the two quantities.

The number of tickets that you sell is measured in tickets and the total amount of money made from ticket sales is measured in dollars. In the second row of the table, write the units used to measure each quantity.

The problem states that the tickets are sold for $2 each and your friend has already sold 100 tickets. Use a variable, such as [[x ]], to represent the number of tickets that you sell. You can determine the total amount of money made from ticket sales by multiplying the total number of tickets sold, [[100+x ]], by the cost of each ticket, 2. The expression [[ 2100+x ]] represents the total amount of money made from ticket sales if you sell [[x ]] tickets. The variable [[x ]] represents the independent quantity and the expression [[ 2100+x ]]represents the dependent quantity. In the third row, write the variable and expression. Quantity name: | Number of Tickets that You Sell | Total Amount of Money Made from Ticket Sales | Units: | tickets | dollars | Expression: | [[x ]] | [[ 2100+x ]] |

Solve a Problem Using the Distributive Property over Multiplication

Example:

Consider the scenario from the previous examples and the table:

You and your friend are selling raffle tickets for the local rotary club. Tickets are sold for $2 each. Your friend has already sold 100 tickets, and you are going to sell the remaining tickets. Quantity name: | Number of Tickets that You Sell | Total Amount of Money Made from Ticket Sales | Units: | tickets | dollars | Expression: | [[x ]] | [[ 2100+x ]] |

How many tickets did you sell if the ticket sales are $464?

You are given the total amount of money made from ticket sales as $464. This is a dependent quantity. You want to determine the number of tickets that you sold if the total amount of money made from ticket sales is $464. Set the expression [[ 2100+x ]] equal to 464 and use the distributive property to solve for [[x ]] to calculate the number of tickets. | [[ 2100+x ]] | = | 464 | | [[ 2100+ 2x ]] | = | 464 | | 200 + [[ 2x ]] | = | 464 | | [[ 2x ]] | = | 264 | | [[x ]] | = | 132 |

Write an Expression Using the Distributive Property over Division

Some expressions representing real-world situations can be easily written in the form [[ ax+b c ]]. When solving problems with expressions written in this form, you will use the distributive property over division.

Example:

Five servers work in a restaurant. Each day, they combine their tips and divide them among themselves evenly. So far today, they have received a total of $45 in tips.

Write an expression for each server's share.

In this problem, you are comparing two quantities: the amount of additional tips and each server's share. In the first row of the table, write the column headings to describe the two quantities.

The amount of additional tips is measured in dollars and each server's share is measured in dollars. In the second row of the table, write the units used to measure each quantity.

The problem states that the five servers divide the total amount in tips evenly and they already have received a total of $45 in tips. Use a variable, such as [[x ]], to represent the amount of additional tips that they receive. You can determine each server's share by dividing the total amount received in tips, [[45+x ]], by the number of servers, 5. The expression [[45+x 5 ]] represents each server's share if they receive [[x ]]more dollars in tips. The variable [[x ]] represents the independent quantity and the expression [[45+x 5 ]]represents the dependent quantity. In the third row, write the variable and expression. Quantity name: | Amount of Additional Tips | Each Server's Share | Units: | dollars | dollars | Expression: | [[x ]] | [[45+x 5 ]] |

Solve a Problem Using the Distributive Property over Division

Example:

Consider the scenario from the previous examples and the table:

Five servers work in a restaurant. Each day, they combine their tips and divide them among themselves evenly. So far today, they have received a total of $45 in tips. Quantity name: | Amount of Additional Tips | Each Server's Share | Units: | dollars | dollars | Expression: | [[x ]] | [[45+x 5 ]] |

What was the amount of additional tips if each server's share was $48?

You are given that each server's share was $48. This is a dependent quantity. You want to determine the amount of additional tips if each server's share was $48. Set the expression [[45+x 5 ]] equal to 48 and use the distributive property to solve for [[x ]] to calculate the amount of additional tips. | [[45+x 5 ]] | = | 48 | | [[45 5+x 5 ]] | = | 48 | | [[9+x 5 ]] | = | 48 | | [[x 5 ]] | = | 39 | | [[x ]] | = | 195 |

So, if each server's share was $48, then the amount of additional tips was $195.

Check For Understanding Which expression shown is written in such a way that the distributive property of multiplication can be used to solve it? * [[ ax+ bx ]] * [[ a bx+c ]] * [[ ax+b c ]] * [[ a bx+c ]]

Expressions written in the form [[ a bx+c ]] can be solved using the distributive property over multiplication.

Your class is collecting canned goods for a food bank. The collection will go on for one school week. Each student brings in 1 can each day and your teacher brings in 10 cans each day. Which expression below could be used to determine the total number of cans collected based on the number of students in your class? * [[ 5x+10 ]] * [[ x5+10 ]] * [[5+x+10 ]] * [[ 5x+10 ]]

Each day, each student brings in 1 can. This is represented by [[ 1x ]], or just [[x ]]. This value is added to the 10 cans the teacher brings in. Finally, this expression must be multiplied through by 5 since the collection goes on for one school week, or 5 days.

Which scenario below could be solved using the distributive property over division? * Four days a week, Tina runs on the track. She takes 5 minutes to stretch then runs the rest of the time. This week she ran a total of 140 minutes. How long did she run each day? * Over summer vacation, Zack reads 2 books a month. If summer vacation lasts for [[x ]]months, how many books does Zack read over the summer? * You and two friends are starting a lawn care business. The three of you will split your total income evenly. So far this month you have made $75 dollars. At the end of the month you each made $375. How much money did you make the rest of the month? * Jonathan is making snack bags for an overnight camping trip. He has 45 carrot sticks. What is the greatest number of bags he can make and still have the same number of carrot sticks in each bag?

In order to use the division, there needs to be a total amount split evenly. To use the distributive property, part of that total amount is unknown.

Lydia's teacher reads the first 30 pages of a novel outloud to her class. She then tells the students they need to finish the novel in the next 5 days. She doesn't tell them how long the novel is but tells them that in order to finish it on time, they have to read 34 pages each day. Lydia solves the following equation to determine the total number of pages in the book. [[x−30 5 ]] | = | 34 | [[x−30 5 ]] | = | 34 | [[x−6 ]] | = | 34 | [[x−6+6 ]] | = | [[34+6 ]] | [[x ]] | = | 40 |

Lydia thinks there are 40 pages in the novel. Which statement below is true about Lydia's work? * Lydia is correct. There are 40 pages in the novel. * Lydia is not correct. She should have used the distributive property of multiplication instead of division. * Lydia is not correct. She did not use the distributive property correctly. * Lydia is not correct. She should have added the 30 pages her teacher read to the total pages to get 70 total pages.

Lydia did use a correct equation but she forgot to distribute the 5 to both the [[x ]] and the 30; thus giving her an answer that does not make sense for this situation.

Additional Resources * Check for Understanding

SKILLS YOU WILL LEARN * Identify quantities and units. * Write an expression. * Determine the value of an independent variable.