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Problem Solving: Using Systems Objective: To Use systems of equations to solve problems

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example Your wireless plan charges.15 a minute during the day and.05 a minute after 9 pm. If your bill for 440 minutes cost 56 dollars how many minutes were used before 9 pm and after 9 pm?

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Step 1 The problem asks for the number of minutes charged at each rate.

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Step 2 Let d be daytime minutes and n be night time minutes

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d + n = 440.15d +.05n = 56 Your wireless plan charges.15 a minute during the day and.05 a minute after 9 pm. If your bill for 440 minutes cost 56 dollars how many minutes were used before 9 pm and after 9 pm? Step 3 set up a system of equations

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Step 4 solve the system d + n = 440.15d +.05n = 56 d + n – n = 440 – n.15d +.05n = 56 d = 440 – n.15d +.05n = 56.15(440 – n) +.05n = 56

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Step 4 continued.15(440 – n) +.05n = 56 distribute 66 –.15n +.05n = 56 Simplify 66 –.10n = 56 66 –.10n – 66 = 56 – 66 -.10n = -10 -.10n/-.10 = -10/-.10 n = 100 minutes

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Step 4 continued If n = 100 and d + n = 440 d + 100 = 440 d + 100 – 100 = 440 – 100 d = 340 Solution: 100 night time minutes and 340 day time minutes

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Step 5 check your answer d + n = 440.15d +.05n = 56 100 + 340 = 440.15(100) +.05(340) = 56

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Try This! A canoe is going up a river and takes 2 hours to travel 30 miles against the current. The trip back takes 1 hour. How fast is the current of the stream? How fast would the canoe move in still water? Hint d = rt The stream is moving at 7½ mph The canoe is moving at 22½ mph

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Solution Let c = canoe speed and s = the streams current c – s is the speed against the current for 2 h. c + s is the speed with the current for 1 h d = 30 d = rt 30 = (c – s)2 30 = (c + s)1

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30 = 2c – 2s distribute 30 = c + s 30 = 2c – 2s multiply bottom equation by 2 (30 = c + s)2to eliminate s. 30 = 2c – 2s add the two equations 60 = 2c + 2s 90 = 4c

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90/4 = 4c/4 22.5 = c 30 = c + s 30 = 22.5 + s substitution 30 – 22.5 = 22.5 + s – 22.5 7.5 = s

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Distance-Rate-Time Applications Example 1: Amy rides her bike to work in 30 minutes. On the way home she catches a ride with a friend and arrives home.

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