Activities

1. How many solutions do the following equations or systems of equations have?
   1. y = 2x                          b. y = x + 3                              c. y = 5x                      d. y = 3
                                             y = 2x                                      y = 5x + 1                    y = 1

2. A 2-liter bottle of soda costs $1.50. Meanwhile, a 2-liter bottle of water costs $1.00. Trisha bought a total of twenty 2-liter bottles for $22.50. You solved this system of equations on a graphing calculator and determined the solution of (15, 5). How many bottles of water did Trisha buy?

3. The system of equations 75b + 150a = 675 and b + a = 7 represents Stuart’s purchase of 7 total bananas (b) and apples (a). Stuart knows that each apple has 150 calories and each banana has 75 calories. How many apples did Stuart buy?

4. The solution to a system of equations is in the form (c, a) where c is the number of cows that can feed comfortably inside an area that is a acres in size. Jerry solved the system, finding the values 112.5 and 61. Which answer makes more sense (61, 112.5) or (112.5, 61) and why?

5. The system of equations w = 1.5m + 3 and w = m + 4.5 describes the weight (w) of two dogs, after a number of months (m) since birth. What does the solution (3, 7.5) represent in this situation?

6. The distance of two horses as a function of their time traveling along a dirt road is graphed on the table below. Estimate the time and distance when they will be at the same location.

7. Tim bought six plain and nine fancy T-shirts at a yard sale for $138. The next week he bought eleven plain and two fancy T-shirts for $108. How much more expensive is one fancy T-shirt than one plain T-shirt?

8. One rental car costs $20 per day and $1.00 per mile. Another rental car costs only $10 per day, but $1.50 per mile. The system of equations c = m + 20 and c = 1.5m + 10 can be used to represent this situation. What is the meaning of the solution (20, 40) to this system of equations?

9. Two objects change their temperature according to the equations t = 55 + 2m and t = 81 – 3m where t is the temperature in degrees Fahrenheit after m minutes of being placed in the same room. At what temperature will the objects reach thermal equilibrium (the same temperature)?

10. Profits (P) for two musicians who are on tour together can be described by the equations P = 4a + 100000 and P = 2a + 100000. They are both guaranteed $100,000 for the tour, while one makes $4 per album sale and the other makes only $2 per album sale. Determine the number of albums they would have to sell to make the same profit.

11. Byron and Maddy grew plenty of organic vegetables during the spring at their garden. They have an overabundance of squash and tomatoes so they decide to sell them at two separate farmers markets. Byron sells 35 tomatoes and 20 squashes for $32.50. Meanwhile, Maddy sells 60 tomatoes and 35 squashes for $56.25. How much money would you have to pay for 4 tomatoes and 2 squashes?

12. Jackie is moving to a new house in another part of her hometown. She purchases a total of 27 small and large boxes to pack her things into. Each small box has a volume of 1500 in³ and each large box has a volume of 4000 in³. If Jackie’s belongings have a volume of 88,000 in³, how many more large boxes does she need than small boxes?

13. Determine a set of lines that has (a) one solution, (b) no solutions, and (c) infinite solutions.

14. An NBA basketball player had a terrific night of scoring last winter. He scored 61 points by making only 26 baskets. Surprisingly, all of his shots were either two or three pointers. How many more points would he have made if all of the baskets were three pointers?

15. The number of tricycles at a repair shop is five less than half the number of bicycles. If there are 118 wheels at the shop, how many more bikes are there than trikes?

16. The sum of two numbers is a, and the difference between two numbers is b. The graph below displays this situation. What are the values of a and b?

17. Two runners start at different points on a dirt road that runs north and south. Using the distance versus time graph below, (assuming that north is positive) describe the differences between the runners starting location, speed, and how far they are from their starting point when they meet.

18. Two barrels are both filled with water from a recent rainy day. In order to not attract misquotes, Trevor wants to put drain holes in each barrel. The equations w = 40 – 2m and w = 55 – 2.5m describe the amount of water (w) in gallons remaining in each barrel after m minutes of draining. Why does the solution (30, -20) not make sense in this situation?

19. Create a system of equations that describes the perimeter and area of a square with side length s being equal. Determine at least one solution to this system. Can there be any other solutions?

20. The sum of the digits of a two-digit number is ten. When the digits are reversed, the number is decreased by 36. What is the original number?

21. Create a story that would use the following system of equations. What would the solution be in your story?

3x + 7y = 27
4x + 13y = 47

22. The system of equations x = 3y + 2 and x = 5y + 2.5 can be used to determine the weight of two dogs based on how old they are. What does the x and y variable represent in this problem? Explain what the solution means in the context of this problem.

23. The zookeepers at a local zoo have been monitoring the weight of two black bears. The first bear, a cub, was 81 lbs. in May and is putting on 5 pounds of weight per month. The second bear, an older female, was 147 lbs. in July and is losing 3 pounds of weight per month. If they continue to change their weights at the same rate, during what month will both bears weigh the same?

24. The solution for the system of equations below is (4, 3). If lines A and B are perpendicular and the y-intercept of line A is 2, what is the y-intercept of line B?

25. Marcus is building a walkway from his house to his shed. He has small and large pavers to use. If Marcus lays down 5 small pavers and 2 large pavers, he can cover 24 inches of distance. On the other hand, if Marcus lays down 2 small pavers and 6 large pavers, he can cover 46 inches of distance. Write the system of equations described by this problem. Then determine the solution and explain what the solution to the system means in this situation.

Answer Key/Rubric

1. a. infinite solutions, b. one solution, c. no solutions, d. no solutions
2. 15 bottles of water
3. 2 apples
4.  (61, 112.5) makes more sense because you cannot have half a cow, but you can have half an acre
5. Both of the dogs will weigh the same, 7.5 lbs. at the same time, 3 months
6. Various answers close to 4.5 hours and 75 miles
7. Fancy T-shirts are $2.00 more than plain T-shirts
8. Both cars will cost the same amount $40.00 after 20 miles of driving
9. 65.4 ^oF
10. If they both sold 0 albums they would have the same profit ($100,000)
11. Four tomatoes and two squash cost $3.50
12. 11 more large packing boxes than small boxes (19, 8)
13. Various answers including: (a) B & C (b) A & D and (c) A & B
14. 17 points
15. 24 more bikes than trikes
16. a = 8 and b = -1
17. Runner A starts further North and runs south faster than runner B who runs much slower. Because of the difference in pace, they meet closer to runner B’s starting point. In fact, runner A covered more than twice the distance that runner B covered.
18. The solution (30, -20) does not make sense in this situation because you cannot have -20 gallons of water in a barrel with holes in the bottom.
19. System of equations P = 4s, A = s², There can not be other solutions
20. 73
21. Various answers including: Tickets were being sold at two rates for a school play. The early bird rate was $3.00 for a student and $7.00 for an adult. They sold $27.00 worth of tickets. The regular price tickets were $4.00 for students and $13.00 for adults. They sold $47.00 worth of tickets. The solution (2, 3) represents the number of student tickets and adult tickets they sold during each type of ticket sale.
22. The variable x represents the weight of the dogs based on y, the dogs age. The solution represents the time when the dogs weigh exactly the same.
23. February
24. b = 19
25. 5s + 2L = 24 and 2s + 6L = 46; The solution is (2, 7) this means that the small pavers cover 2 inches and the large pavers cover 7 inches