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2.5 EQUATIONS AND PROBLEM SOLVING

2.5 EQUATIONS AND PROBLEM SOLVING. Do now:. The sum of two consecutive even integers is 118. Find the integers. The sum of two consecutive odd integers is 56. find the integers. Objectives: To define a variable in terms of another variable To model distance-rate-time problems Vocabulary:

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2.5 EQUATIONS AND PROBLEM SOLVING

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  1. 2.5 EQUATIONS AND PROBLEM SOLVING

  2. Do now: • The sum of two consecutive even integers is 118. Find the integers. • The sum of two consecutive odd integers is 56. find the integers.

  3. Objectives: • To define a variable in terms of another variable • To model distance-rate-time problems Vocabulary: • Consecutive integers • Uniform motion Standards: 15

  4. Defining one variable in terms of another: Ex. The length of a rectangle is 6 inches more than its width. The perimeter of the rectangle is 24 inches. What is the length of the rectangle.

  5. You do: The length of a rectangle is 8 cm more than twice its width. The perimeter of the rectangle is 34 inches. What is the length of the rectangle.

  6. Distance-rate problems: The formula d = rtgives the relationship between distance d, rate r, and time t. http://www.brainpop.com/math/algebra/distancerateandtime/preview.weml

  7. D = r t What does D = r  t represent? time Distance = rate 

  8. Austin Two trucks left Austin traveling in opposite directions. One truck traveled at a rate of 80 km/h and the other traveled at a rate of 70 km/h. After how many hours were the trucks 300 km apart? D = r  t

  9. D = 70t Austin Two trucks left Austin traveling in opposite directions. One truck traveled at a rate of 80 km/h and the other traveled at a rate of 70 km/h. After how many hours were the trucks 300 km apart? D = r  t

  10. D = 80t D = 70t Austin Two trucks left Austin traveling in opposite directions. One truck traveled at a rate of 80 km/h and the other traveled at a rate of 70 km/h. After how many hours were the trucks 300 km apart? D = r  t

  11. D = 80t D = 70t Austin Two trucks left Austin traveling in opposite directions. One truck traveled at a rate of 80 km/h and the other traveled at a rate of 70 km/h. After how many hours were the trucks 300 km apart? D = r  t

  12. 150 150 Two trucks left Austin traveling in opposite directions. One truck traveled at a rate of 80 km/h and the other traveled at a rate of 70 km/h. After how many hours were the trucks 300 km apart? 70t + 80t = 300 150t = 300 t = 2 hours D = r  t

  13. Atlanta A truck left Atlanta and traveled west at 80 km/h. Another truck left and headed east at 60 km/h. How many hours had the first truck traveled when they were 420 miles apart? D = r  t

  14. D = 80t Atlanta Atruck left Atlanta and traveled west at 80 km/h. Another truck left and headed east at 60 km/h. How many hours had the first truck traveled when they were 420 miles apart? D = r  t

  15. D = 60t D = 80t Atlanta A truck left Atlanta and traveled west at 80 km/h.Another truck left and headed east at 60 km/h. How many hours had the first truck traveled when they were 420 miles apart? D + D = 420 D = r  t

  16. D = 60t D = 80t Atlanta A truck left Atlanta and traveled west at 80 km/h.Another truck left and headed east at 60 km/h. How many hours had the first truck traveled when they were 420 miles apart?

  17. A truck left Atlanta and traveled west at 80 km/h.Another truck left and headed east at 60 km/h. How many hours had the first truck traveled when they were 420 miles apart?

  18. D.C. Maria left Washington and drove to Philadelphia at 40 km/h. Jose left Washington 2 hours later and took the same route to Philadelphia at a rate of 80km/h. If both drivers arrived in Philadelphia at the same time, How long did Maria spend on the road? D = r  t

  19. Phili D = 40t D.C. Maria left Washington and drove to Philadelphia at 40 km/h.Jose left Washington 2 hours later and took the same route to Philadelphia at a rate of 80km/h. If both drivers arrived in Philadelphia at the same time, How long did Maria spend on the road? D = r  t

  20. Phili D = 40t D = 80(t – 2) D.C. Maria left Washington and drove to Philadelphia at 40 km/h.Jose left Washington 2 hours later and took the same route to Philadelphia at a rate of 80km/h. If both drivers arrived in Philadelphia at the same time, How long did Maria spend on the road? D = D

  21. D = 40t D = 80(t – 2) Maria left Washington and drove to Philadelphia at 40 km/h.Jose left Washington 2 hours later and took the same route to Philadelphia at a rate of 80km/h. If both drivers arrived in Philadelphia at the same time, How long did Maria spend on the road? D = D

  22. Phili D = 40t D = 80(t – 2) 40 40 D.C. Maria left Washington and drove to Philadelphia at 40 km/h.Jose left Washington 2 hours later and took the same route to Philadelphia at a rate of 80km/h. If both drivers arrived in Philadelphia at the same time, How long did Jenny spend on the road? D = D 40t = 80(t – 2) 40t = 80t – 160 -40t -40t 0 = 40t – 160 160 = 40t t = 4 hours D = r  t

  23. A train leaves a train station at 1pm. It travels at a constant rate of 72mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 90mi/h. T he second train follows the same route as the first train on a parallel track. In how many hours will the second train catch up with the first train 72mi/h 90mi/hr

  24. A train leaves a train station at 1pm. It travels at a constant rate of 72mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 90mi/h. T he second train follows the same route as the first train on a parallel track. In how many hours will the second train catch up with the first train

  25. A train leaves a train station at 1pm. It travels at a constant rate of 72mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 90mi/h. T he second train follows the same route as the first train on a parallel track. In how many hours will the second train catch up with the first train

  26. A train leaves a train station at 1pm. It travels at a constant rate of 72mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 90mi/h. T he second train follows the same route as the first train on a parallel track. In how many hours will the second train catch up with the first train.

  27. A motorboat took 3 hours to make a downstream trip with a current of 3 mph. The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = r  t

  28. D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph. The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = r  t

  29. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D D = r  t

  30. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D

  31. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D

  32. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D

  33. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D 3(r + 3) = 12(r – 3) 3r + 9 = 12r – 36 -3r -3r 9 = 9r – 36 +36 +36 45 = 9r D = r  t

  34. D = 12(r – 3) D = 3(r + 3) A motorboat took 3 hours to make a downstream trip with a current of 3 mph.The return trip against the same current took 12 hours. Find the speed of the boat in still water. D = D 3(r + 3) = 12(r – 3) 3r + 9 = 12r – 36 -3r -3r 9 = 9r – 36 +36 +36 45 = 9r 9 9 r = 5 mph D = r  t

  35. Ben paddles his kayak upstream in 4 hours, against a current of 1 mile per hour. He turns around and paddles back to where he started in 2 hours. What is Ben’s paddling speed? D = r  t

  36. D = 4(r – 1) Ben paddles his kayakupstream in 4 hours, against a current of 1 mile per hour. He turns around and paddles back to where he started in 2 hours. What is Ben’s paddling speed? D = r  t

  37. D = 4(r – 1) D = 2(r + 1) Ben paddles his kayakupstream in 4 hours, against a current of 1 mile per hour.He turns around and paddles back to where he started in 2 hours.What is Ben’s paddling speed? D = D

  38. D = 4(r – 1) D = 2(r + 1) Ben paddles his kayakupstream in 4 hours, against a current of 1 mile per hour.He turns around and paddles back to where he started in 2 hours.What is Ben’s paddling speed? D = D

  39. D = 4(r – 1) D = 2(r + 1) r r Ben paddles his kayakupstream in 4 hours, against a current of 1 mile per hour.He turns around and paddles back to where he started in 2 hours.What is Ben’s paddling speed? D = D 4(r – 1) = 2(r + 1) 4r – 4 = 2r + 2 -2r -2r 2r – 4 = 2 2r = 6 r = 3 mph D = r  t

  40. Do now: The length of a rectangle is 4 inches more than 3 times its width. The perimeter of the rectangle is 48 inches. What is the length of the rectangle. w L

  41. A passenger plane made a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours?

  42. A passenger plane made a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours?

  43. A cattle train left Miami and traveled toward New York. 14 hours later a diesel train left traveling at 45 km/h in an effort to catch up to the cattle train. After traveling for four hours the diesel train finally caught up. What was the cattle train's average speed?

  44. A cattle train left Miami and traveled toward New York. 14 hours later a diesel train left traveling at 45 km/h in an effort to catch up to the cattle train. After traveling for four hours the diesel train finally caught up. What was the cattle train's average speed?

  45. A cattle train left Miami and traveled toward New York. 14 hours later a diesel train left traveling at 45 km/h in an effort to catch up to the cattle train. After traveling for four hours the diesel train finally caught up. What was the cattle train's average speed? r = 10km/h

  46. A cargo plane flew to the maintenance facility and back. It took one hour less time to get there than it did to get back. The average speed on the trip there was 220mph. The average speed on the way back was 200 mph. How many hours did the trip there take? • Ryan left the science museum and drove south Gabriella left three hours later driving 42 km/h faster in an effort to catch up to him. After two hours Gabriella finally caught up. Find Ryan's average speed. • Kali left school and traveled toward her friend's house at an average speed of 40 km/h. Matt left one hour later and traveled in the opposite direction with an average speed of 50 km/h. Find the number of hours Matt needs to travel before they are 400 km apart.

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