4.2 Equal Values Method Name: Recall Writing and solving equations involving rate of change and initial value a. In a jumping frog race, your frog received a 6 in. head start and jumps 3 in. every 2 seconds. How far will your frog jump in 10 seconds? ROC: IV: Equation: What do the variables represent in your equation? x = y = Solution: b. Your parents put a down payment of $510 on your car but they are requiring you to pay the monthly payment of $85. If the care cost a total of $2125, how long will it take you to pay of the car? ROC: IV: Equation: What do the variables represent in your equation? x = y = Solution: Solving Systems Method #2 Equal Values Equal Values Method - or- Finding the Break Even Point Given the system y = 2x + 5 and y = x 1 find the point of intersection. Since both equations are equal to y, they are equal to each other. 2x + 5 = x 1 This eliminates one of the variables. So, now you can solve for x. 3x + 5 = 1 3x = 6 x = 2 Now, substitute x into one of the original equations and solve for y. y = 2 1 y = 1 Solution (2, 1) Use the Equal Values method to solve each system. a. y = 3x 4 b. y =!! x + 4 y =!! x + 7 y =!! x 2
The Situation- Charles and Amy are each growing a tree. Charles tree is 3 ft. tall when he plants it and is expected to grow 1.5 ft. per year. On the same day, Amy plants a tree seed. The tree from the seed is expected to grow 1.75 ft. per year. Will the trees ever be the same size? If so, how many years will it take? How tall will they be? Solve the problem using a table, a graph and a system of equations. Be sure to use a scale and label your graph. Define the variables: Let x = Let y = Charles equation: Amy s equation: Age (years) Charles Tree Amy s Tree Solve using the equal values method. What is the domain and range of Charles function? What is the domain and range of Amy s functions?
Example 1: Herman and Jackie are saving money to pay for college. Herman currently has $15,000 and is working hard to save $1000 per month. Jackie only has $12,000 but is saving $1300 per month. In how many months will they have the same amount of savings? How much will each of them have saved? Let x = Let y = Write an equation for Herman s savings. Write an equation for Jackie s savings. In how many months will they have the same amount of savings? How much will each of them have saved? Example 2: Scientists studied the weights of two alligators over a period of 12 months. The initial weight and growth rate of each alligator is shown in the table. After how many months did the alligators weigh the same? Initial Weight (lbs) Growth Rate (lbs) Alligator 1 4 1.5 Alligator 2 6 1 Let x = Let y = Write an equation Alligator 1: Alligator 2: After how many months did the alligators weigh the same? How much did they weight?
Practice: Solve each problem using the Equal Values method. 1. y = 2x 3 2. y = 3x + 11 3. x + y = 5 x + y = 15 x + y = 3 2y x = 2 4. x + 2y = 10 5. x + 2y = 4 6. 3x = y 2 3x 2y = 2 x + 2y = 6 6x + 4 = 2y 7. Wendy is starting a catering business and is attempting to figure out who she should hire to transport the food to different locations. She has found two trucking companies. Peter s Pick Up charges $0.40 per mile and charges a flat fee of$68. Helen s Haulers charges $0.65 per mile and charges a flat fee of $23. For what distance would the cost of transporting to the produce be the same for both companies? What is that equal cost? Write a system of equations to model the situation. Solve and answer the questions. For what distance would the cost of transporting to the produce be the same for both companies? What is that equal cost? Use your equations to compare the companies: Which company charges a lower fee for a 160 mile trip? Which company will move a greater distance for $200?
8. Jonas needs a cell phone. He has a choice between two companies with the following monthly billing policies. Each company s monthly billing policy has an initial operating fee and charge per minute. Company Operating Fee Charge per Minute Terry s Telephone 29.95 0.l4 Carrie s Connections 4.95 0.39 At how many minutes is the monthly cost the same? What is the equal monthly cost of the two plans for those minutes? 9. George bought some CDs at his local store. He paid $15.95 for each CD. Nora bought the same number of CDs from a store online. She paid $13.95 for each CD, but had to pay $8 for shipping. In the end, both George and Nora spent the exact same amount of money buying their CDs! How many CDs did George buy? How much did each of them spend? How many CDs did George buy? How much did each of them spend?
10. Movies- Are- Us has two video rental plans. The Regular video rental plan charges $3.25 for each video rental. The Preferred video rental plan has an $ 8.75 membership fee and charges $ 2 for each video rental. How many video rentals give the two plans the same cost? What is the equal cost? How many video rentals give the two plans the same cost? What is the equal cost? Use your equations to compare the plans: Which video plan costs more for 18 video rentals? Which plan provides more videos for $ 30.00? 11. The sales of Gizmo Sports Drink at the local supermarket are currently 6,500 bottles per month. Since New Age Refreshers were introduced, sales of Gizmo have been declining by 55 bottles per month. New Age currently sells 2,200 bottles per month and its sales are increasing by 250 bottles per month. If these rates of change remain the same, in about how many months will the sales for both companies be the same? How many bottles will each company be selling at that time? How many months will the sales for both companies be the same? How many bottles will each company be selling at that time?