Algebra 2: Systems of Equations Topic D5: Applications of Systems of Equations | VERSION A www.varsitylearning.com 1 Name: _________________________________________________ 1. One number is 2 more than 2 times another. The sum of the numbers is 11. Find the two numbers. The two numbers are and . 2. The sum of three numbers is 11. Two times the smallest is 2 less than the largest, while the sum of the largest and smallest is 8. Use a linear system in three variables to find the three numbers. The three numbers are , , and . 3. A mixture of 18% disinfectant solution is to be made from 16% and 21% disinfectant solution. How much of each solution should be used if 10 gallons of the 18% solution are needed? There needs to be gallons of 16% alcohol solution and gallons of 21% solution. 4. A total of 775 tickets were sold for a game for a total of $1,350.00. If adult tickets sold for $2.00 and children's tickets sold for $1.50, how many of each kind of tickets were sold? They sold adult tickets and children's tickets. 5. Susan invests 2 times as much money at 9% as she does at 5%. If her total interest after 1 year is $1,380, how much does she have invested at each rate? Susan invests $ at 5% and $ at 9%. 6. How many gallons of 25% alcohol solution and 50% alcohol solution must be mixed to get 10 gallons of 30% alcohol solution? There needs to be gallons of 25% alcohol solution and gallons of 50% solution. 7. The difference of two positive numbers is 10. The larger number is fifteen less than twice the smaller number. Find the numbers. The numbers are , and . 8. A chemist has three different acid solutions. The first acid solution contains 20% acid, the second contains 35% and the third contains 75%. He wants to use all three solutions to obtain a mixture of 306 liters containing 30% acid, using 3 times as much of the 75% solution as the 35% solution. How many liters of each solution should be used? The chemist should use liters of 20% solution, liters of 35% solution, and liters of 75% solution.
Transcript
Algebra 2: Systems of Equations
Topic D5: Applications of Systems of Equations | VERSION A
1. One number is 2 more than 2 times another. The sum of the numbers is 11. Find the two numbers. The two numbers are and .
2. The sum of three numbers is 11. Two times the smallest is 2 less than the largest, while the sum of the largest and smallest is 8. Use a linear system in three variables to find the three numbers. The three numbers are , , and .
3. A mixture of 18% disinfectant solution is to be made from 16% and 21% disinfectant solution. How much of each solution should be used if 10 gallons of the 18% solution are needed? There needs to be gallons of 16% alcohol solution and gallons of 21% solution.
4. A total of 775 tickets were sold for a game for a total of $1,350.00. If adult tickets sold for $2.00 and children's tickets sold for $1.50, how many of each kind of tickets were sold? They sold adult tickets and children's tickets.
5. Susan invests 2 times as much money at 9% as she does at 5%. If her total interest after 1 year is $1,380, how much does she have invested at each rate? Susan invests $ at 5% and $ at 9%.
6. How many gallons of 25% alcohol solution and 50% alcohol solution must be mixed to get 10 gallons of 30% alcohol solution? There needs to be gallons of 25% alcohol solution and gallons of 50% solution.
7. The difference of two positive numbers is 10. The larger number is fifteen less than twice the smaller number. Find the numbers. The numbers are , and .
8. A chemist has three different acid solutions. The first acid solution contains 20% acid, the second contains 35% and the third contains 75%. He wants to use all three solutions to obtain a mixture of 306 liters containing 30% acid, using 3 times as much of the 75% solution as the 35% solution. How many liters of each solution should be used? The chemist should use liters of 20% solution, liters of 35% solution, and liters of 75% solution.
Algebra 2: Systems of Equations
Topic D5: Applications of Systems of Equations | VERSION A
9. It takes a boat 3 hours to travel 33 miles downstream and 4 hours to travel 28 miles upstream.
a. What is the speed of the boat in still water?
b. What is the speed of the current of the river?
10. An airplane flying with the wind can cover 620 miles
in 2 hours. The return trip against the wind takes 2!!
hours.
a. What is the speed of the plane?
b. What is the speed of the wind?
11. Bob has 16 coins totaling $1.20. If he only has dimes and nickels, how many of each coin does he have? Bob has nickels and dimes.
12. A collection of nickels, dimes, and quarters consist of 65 coins with a total of $7.75. If there are 2 times as many dimes as quarters, find the number of each type of coins. There are nickels, dimes and quarters.
13. A manufacturing company finds that they can sell 425 items at $1.50 per item and 300 items at $2.00 per item. If the relationship between the number of items sold 𝑥 and the price per item 𝑝 is a linear one:
a. Find the formula that gives 𝑥 in terms of 𝑝: 𝑥 =
b. Now use the formula to find the number of items they will sell if the price per item is $3.00. They will sell items if the price is $3.00.
14. A ball is tossed into the air so that the height after 1, 2, and 5 seconds is as given in the following table. If the relationship between the height of the ball, ℎ, and the time, 𝑡, is quadratic, then the relationship can be written as: ℎ = 𝑎𝑡! + 𝑏𝑡 + 𝑐. Use the information in the table to write a system of three equations in three variables 𝑎, 𝑏, and 𝑐. Solve the system to find the exact relationship between ℎ and 𝑡.
Time 1 sec 2 sec 5 sec
Height 128 m 144 m 0
𝑎 =
𝑏 =
𝑐 =
Algebra 2: Systems of Equations
Topic D5: Applications of Systems of Equations | VERSION A
1. One number is 2 more than 2 times another. The sum of the numbers is 11. Find the two numbers. The two numbers are 3 and 8 .
2. The sum of three numbers is 11. Two times the smallest is 2 less than the largest, while the sum of the largest and smallest is 8. Use a linear system in three variables to find the three numbers. The three numbers are 2 , 6 , and 3 .
3. A mixture of 18% disinfectant solution is to be made from 16% and 21% disinfectant solution. How much of each solution should be used if 10 gallons of the 18% solution are needed? There needs to be 6 gallons of 16% alcohol solution and 4 gallons of 21% solution.
4. A total of 775 tickets were sold for a game for a total of $1,350.00. If adult tickets sold for $2.00 and children's tickets sold for $1.50, how many of each kind of tickets were sold? They sold 375 adult tickets and 400 children's tickets.
5. Susan invests 2 times as much money at 9% as she does at 5%. If her total interest after 1 year is $1,380, how much does she have invested at each rate? Susan invests $ 6,000 at 5% and $ 12,000 at 9%.
6. How many gallons of 25% alcohol solution and 50% alcohol solution must be mixed to get 10 gallons of 30% alcohol solution? There needs to be 8 gallons of 25% alcohol solution and 2 gallons of 50% solution.
7. The difference of two positive numbers is 10. The larger number is fifteen less than twice the smaller number. Find the numbers. The numbers are 25 , and 35 .
8. A chemist has three different acid solutions. The first acid solution contains 20% acid, the second contains 35% and the third contains 75%. He wants to use all three solutions to obtain a mixture of 306 liters containing 30% acid, using 3 times as much of the 75% solution as the 35% solution. How many liters of each solution should be used? The chemist should use 238 liters of 20% solution, 17 liters of 35% solution, and 51 liters of 75% solution.
Algebra 2: Systems of Equations
Topic D5: Applications of Systems of Equations | VERSION A
9. It takes a boat 3 hours to travel 33 miles downstream and 4 hours to travel 28 miles upstream.
a. What is the speed of the boat in still water? 9 mph
b. What is the speed of the current of the river? 2 mph
10. An airplane flying with the wind can cover 620 miles
in 2 hours. The return trip against the wind takes 2!!
hours.
a. What is the speed of the plane? 279 mph
b. What is the speed of the wind? 31 mph
11. Bob has 16 coins totaling $1.20. If he only has dimes and nickels, how many of each coin does he have? Bob has 8 nickels and 8 dimes.
12. A collection of nickels, dimes, and quarters consist of 65 coins with a total of $7.75. If there are 2 times as many dimes as quarters, find the number of each type of coins. There are 20 nickels, 30 dimes and 15 quarters.
13. A manufacturing company finds that they can sell 425 items at $1.50 per item and 300 items at $2.00 per item. If the relationship between the number of items sold 𝑥 and the price per item 𝑝 is a linear one:
a. Find the formula that gives 𝑥 in terms of 𝑝: 𝑥 = −250𝑝 + 800
b. Now use the formula to find the number of items they will sell if the price per item is $3.00. They will sell 50 items if the price is $3.00.
14. A ball is tossed into the air so that the height after 1, 2, and 5 seconds is as given in the following table. If the relationship between the height of the ball, ℎ, and the time, 𝑡, is quadratic, then the relationship can be written as: ℎ = 𝑎𝑡! + 𝑏𝑡 + 𝑐. Use the information in the table to write a system of three equations in three variables 𝑎, 𝑏, and 𝑐. Solve the system to find the exact relationship between ℎ and 𝑡.
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