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Aim: How do we solve verbal problems using two variables?. Do Now:.

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## Aim: How do we solve verbal problems using two variables?

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**Aim: How do we solve verbal problems using two variables?**Do Now: Jonathan left his home by car, traveling on a certain road at the rate of 45 mph. Three hours later, his brother Jessie left the home and started after him on the same road, traveling at a rate of 60 mph. In how many hours did Jessie overtake Jonathan?**Jonathan**Jessie 60h = 45(h + 3) 60h = 45(h + 3) 60(9) = 45(9 + 3) 540 = 540 Do Now D = rt Jonathan traveled 3 hours longer 45(h + 3) 60h h = the number of hours Jessie traveled. h + 3= hours traveled by Jonathan 60h = the distance traveled by Jessie 45(h + 3) = the distance traveled by Jonathan 60h = 45h + 135 15h = 135 h = 9**Using Two Variables**The larger of two numbers is 4 times the smaller. If the larger number exceeds the smaller by 15, find the number. Let x= smaller # Let 4x = larger # 4x = x + 15 3x = 15 x = 5 4x = 20 Use a system of equations to solve the same problem Let x = smaller # Let y = larger # y = 4x x = y - 15 Substitution x= 4x - 15 -3x = -15 y = 4x = 4(5) = 20 x = 5**x + y = 8.6**2x + 2y = 17.2 -2x + 3y = 6.3 3y – 2x = 6.3 x + y = 8.6 x + 4.7 = 8.6 x = 3.9 Model Problem Use a system of equations to solve The sum of two numbers is 8.6. Three times the larger number decreased by twice the smaller is 6.3. What are the numbers? Let x = smaller # Let y = larger # x + y = 8.6 Additive inverse 3y – 2x = 6.3 2(x + y = 8.6) 5y = 23.5 y = 4.7 The two numbers are 4.7 and 3.9**60h = 45t**Do Now – Two Variables Jonathan left his home by car, traveling on a certain road at the rate of 45 mph. Three hours later, his brother Jessie left the home and started after him on the same road, traveling at a rate of 60 mph. In how many hours did Jessie overtake Jonathan? h = the number of hours Jessie traveled. t = hours traveled by Jonathan 60h = the distance traveled by Jessie 45t = the distance traveled by Jonathan h + 3 = t 60h = 45(h + 3) 15h = 135 h = 9**Aim: How do we solve verbal problems using two variables?**Do Now: Mario had $6.50, consisting of dimes and quarters, in a coin bank. The number of quarters was 10 less than twice the number of dimes. How many coins of each kind did he have?**Use a system of equations to solve the same problem.**Let d= # of dimes q = # of quarters .10d= value of dimes .25q = value of quarters .10d + .25q = 6.50 10d + 25q = 650 q = 2d - 10 Substitution 10d + 25(2d – 10) = 650 10d + 50d – 250 = 650 60d – 250 = 650 60d = 900 d = 15 q = 2d – 10 = 2(15) – 10 = 20 15 dimes = $1.50 20 quarters = $5.00 1.50 + 5.00 = $6.50**6b + 8h = 140**3(6b + 8h = 140) 9b + 6h = 132 -4(9b + 6h = 132) 18b + 24h = 420 -36b - 24h = -528 6(6) + 8h = 140 36 + 8h = 140 8h = 104 h = 13 Model Problem The owner of a men’s clothing store bought six belts and eight hats for $140. A week later, at the same prices, he bought nine belts and six hats for $132. Find the price of a belt and the price of a hat. Let b = belt Let h = hat 6b + 8h = 140 Additive inverse - eliminate h 9b + 6h = 132 -18b = -108 b = 6 The belts costs $6 ea. And the hats cost $13 ea.**Use a system of equations to solve**A dealer wishes to obtain 50 pounds of mixed cookies to sell for $3.00 per pound. If he mixes cookies worth $3.60 per pound with cookies worth $2.10 per pound, find the number of pounds of each kind he should use. Let x = #lb. of cookie 1 Let y = #lb. of cookie 2 Value of cookie #1 Value of cookie #2 3.60x 2.10y x + y = 50 Substitution 3.60x + 2.10y = 150 x = 50 - y 360x + 210y = 15000 360(50 – y) + 210y = 15000 18000 – 360y + 210y = 15000 – 150y = -3000 y = 20 lb. of cookie 1 - $2.10 x + y = 50 x + 20 = 50 x = 30 lb. of cookie 2 - $3.60**20mph**12mph Use a system of equations to solve A motor boat can travel 60 miles downstream in 3 hours. It requires 5 hours to make the return trip against the current. Find the rate of the boat in still water and the rate of the current. Let r = boat’s rate in still water Let c = current’s rate r + c = boat’s rate going downstream r - c = boat’s rate going upstream Additive inverse - eliminate c r + c = 20 r - c = 12 2r = 32 r = 16mph rate of boat in still water r + c = 20 16 + c = 20 c = 4mph rate of current

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