Lesson 5.7 Problem Solving Using Linear Models

Formulas to use:

y = mx + b / This is used when you have a constant rate of change (m)
Ax + By = C / This is used when you have two variables that have a sum that is a constant.

Example 1:

A salesclerk at a clothing store, Julie, receives a monthly base pay of $800 plus a 5% commission on the amount she sells. Write a linear model that gives Julie’s total monthly pay in terms of the amount of clothes she sells. Then find Julie’s monthly pay if she sold $4000 in clothing.

Step 1: verbal model

Total Pay / = / Commission Rate / ∙ / Amount Sold / + / Base Pay

Step 2: Substitute in known values

y / = / 0.05 / ∙ / x / + / 800

This is your linear model. Y = 0.05x + 800. You can use this equation to find how much Julie made during a month when you know she sold $4000 in clothing.

Step 3: Use the equation to find an unknown value.

y = 0.05(4000) + 800

y = $1000

Example 2

The school math club decides to have a fund-raising project. The club rents the movie “The Life and Times of René Descartes” for $19.75, and charge an admission fee of $0.25 per person. Write an equation that gives the profit, P, in terms of the number of people, n, who attend the movie. After counting the receipts, the club members found that they earned a profit of $40.75. How many people attended the movie?


Example 3

At Charles Drew High School, the main items on the lunch menu are often hamburgers and French fries. State law requires that school lunches contain at least 850 calories. The hamburger bun has 80 calories, a container of milk has 160 calories, the ground beef has 85 calories per ounce, and French fries have 70 calories per ounce. Write a linear equation that relates the number of ounces of ground beef, x, and the number of ounces of fries, y, that can be used to create a lunch consisting of 850 calories. If the hamburger is 4.5 ounces, how many ounces of French fries can a student receive?

Step 1: verbal model

Ground Beef / + / French fries / + / Hamburger Bun / + / Milk / = / 850 Calories

Step 2: Substitute in known values

85x / + / 70y / + / 80 / + / 160 / = / 850

Simplify the equation:

85x + 70y + 240 = 850

85x + 70y = 610

Step 3: Use the equation to find an unknown value

85(4.5) + 70y = 610

382.5 + 70y = 610

70y = 227.5

y = 3.25

A student can receive 3.25 ounces of French fries.

Example 4

You are buying $48 worth of lawn seed that consists of two types of seed. One type is quick-growing rye grass that costs $4 per pound, and the other type is a higher-quality seed that costs $6 per pound. Write an equation that represents the different amounts of $4 seed, x, and the $6 seed, y, that you could buy. Then find out how many pounds of $4 seed you could buy if you purchased 3.5 pounds of $6 seed.