Problems that occur in business situations often require expressing income as a linear function of one variable like time worked or number of sales. For example, if an employee earns $7.25 per hour, then income I earned for working h hours is given by I = 7.25h. If a movie theater charges $8.50 for admission, then the income R from selling n admission tickets is given by R = 8.50n. Questions about those situations can be expressed as equations or inequalities. For example, the question How many hours does the employee need to work to earn $290? is represented by the equation 290 = 7.25h. The question How many admission tickets does the movie theater need to sell to collect at least $1,000? is represented by the inequality 8.50n 1,000. In many business situations, income is a function of several variables. The income function is built by combining two or more simple linear functions. As you work on the problems of this investigation, look for answers to these questions: How can you use linear functions of two independent variables to represent problem situations? How can you graph and find solutions for linear equations in two variables? 1
1. Many middle and high school students work on a service for hire basis until they reach the minimum age required to apply for a real job. Bret mows lawns and washes cars in his neighborhood to earn spending money. He charges $20 per lawn and $10 per car wash. a. How much money would Bret earn for mowing $300 15 lawns? For washing $200 20 cars? What would be Bret s total income for 15 lawns and 20 car $500 washes? b. Bret s total income I is a function of the numbers of lawns mowed L and cars washed C. Write a rule that expresses I as a function of L and C. I = 20L + 10C c. Suppose that Bret has a goal to earn $1,200 and has scheduled 50 lawn mowing jobs for the summer. How many cars must Bret wash to reach the income goal? What if Bret only schedules 40 lawn mowing jobs? I = 20L + 10C 1200 = 20(50) + 10C 1200 = 1000 + 10C 200 = 10C 20 = C car washes I = 20L + 10C 1200 = 20(40) + 10C 1200 = 800 + 10C 400 = 10C 40 = C car washes 2
2 Suppose that Bret sets an income goal of $2,000. a. Write an equation that represents the question How many lawn mowings and how many car washes will it take to achieve Bret s income goal? b. If Bret only mows lawns, how many would it take to reach the income goal? c. How many cars would Bret have to wash to meet the income goal by car washing only? d. Find 2 more combinations of number of lawn mowings and number of car washes that would achieve Bret s income goal. Be prepared to explain why these (L, C) combinations are solutions of the equation from Part a. 3
3 Plot the (L, C) pairs that you found to be solutions of 20L + 10C = 2,000 on a coordinate grid like that pictured below. Identify coordinates of two other points that seem to fit the pattern of the plotted points. Check to see if those pairs of numbers are also solutions of the equation. 4 The graph of (L, C) pairs that will achieve the $2,000 income goal suggests that C is a linear function of L. a. Rewrite the equation 20L + 10C = 2,000 to express C as a linear function of L. b. Write a rule expressing L as a function of C. Is L a linear function of C? Explain why or why not. 4
Consider this problem Your school is sponsoring a pancake dinner to raise money for a field trip. You estimate that 200 adults and 250 children will attend. Let x represent the cost of an adult ticket and y represent the cost of a children s ticket. In your GROUPS, write an equation that can be used to describe the price each ticket should be set at to make $3800. 200x + 250y = 3800 Does anyone know in what form is this equation written? Standard Form Ax + By = C, where A, B, and C are numbers. Why do we use it? It is easy to find x and y intercepts using standard form. x and y intercepts Remember what x and y intercepts are? x intercept: where the graph crosses the x axis. y coordinate = 0 y intercept: where the graph crosses the y axis. x coordinate = 0 MOST IMPORTANTLY, THEY ARE ORDERED PAIRS!!!! How do we find them? To find an x intercept, substitute 0 in for y and solve for x (x, 0) To find a y intercept, substitute in 0 for x and solve for y (0, y) REMEMBER, THEY ARE ORDERED PAIRS (x, y) 5
Find the x and y intercepts x 5y = 20 2x 3y = 6 9x 4y = 16 6
How to find slope The easiest way to find slope is to convert from standard form to slope intercept form. Slope Intercept form: y = mx + b Slope = m It is the coefficient of x (the number multiplied by x) ONLY when the equation is in Slope Intercept Form Use inverse operations to solve for y. Try these! 6x + 2y = 12 2x 5y = 12 7
7 A local radio station sponsors a T shirt toss and floppy hat drop during home pro basketball games. The T shirts cost the radio station $8 each, and the hats cost $12 each. c. Suppose the radio station has budgeted $2,400 per game for giveaways. Write an equation that represents the question How many shirts and hats can the radio station give away for $2,400? e. Rewrite your equation from Part c to express s as a function of h. Explain what the slope and s intercept of this linear function tell you about the situation. 8