Applications of Systems of Equations

Three Steps to solving applications

Step 1: NAME YOUR VARIABLES!! What are you looking for and what are you going to call them in math language?

Step 2: Write the number of equations that equals the number of variables.

Step 3: Line up variables and solve using matrices OR solve for y and graph.

Example 1

You are selling tickets at a high school football game. Student tickets cost $2, and general admission tickets cost $3. You sell 1957 tickets and collect $5035. How many of each type of ticket did you sell?

YOU DO #1 : Which of the following would be your variables?

A. x = # of 3 point questionsB. y = amount of moneyC. x = total # of questionsD. y = # of 5 point questions

You Do #1: Which of the following equations would you use to solve the problem?

A. x + y = 150B. x + y = 46C. 5x + 3y = 150 D. 3x + 5y = 150

What is the solution to YOU DO #1

A. 5 3pt questions and 3 5pt questions

B. 196 3pt questions and 888 5pt questions

C. 40 3pt questions and 6 5pt questions

D. 6 3pt questions and 40 5pt questions

Example 2

A collection of nickels and dimes is worth $3.30. There are 42 coins in all. How many of each coin are there?

YOU DO #2 : Which of the following would be your variables?

A. x = # of coinsB. y = amount of moneyC. x = # of dimesD. y = # of quarters

You Do #2: Which of the following equations would you use to solve the problem?

A. x + y = 100B. x + y = 21.40C. 10x + 25y = 21.40 D. .10x + .25y = 21.40

What is the solution to YOU DO #2

A. 76 quarters and 24 dimesB. 24 dimes and 76 quartersC. 165 dimes and -65 quartersD. 121 dimes and 15 quarters

Example 3

The perimeter of a lot is 84 feet. The length exceeds the width by 16 feet. Find the length and width of the lot.

YOU DO #3 : Which of the following would be your variables?

A. x = perimeterB. y = widthC. x = lengthD. y = dimensions

You Do #3: Which of the following equations would you use to solve the problem?

A. x + y = 42B. 2x + 2y + 42C. x = y + 4 D. y = x +4

What is the solution to YOU DO #3

A. 12.5 ft long X 8.5 ft wideB. 23 ft long X 19 ft wideC. 12.5 ft wide X 8.5 ft longD. 23 ft wide X 23 ft long

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Example 4

Three solutions contain a certain acid. The first contains 10% acid, the second 30%, and the third 50%. A chemist wishes to use all three solutions to obtain a 50 – liter mixture containing 32% acid. If the chemist wants to use twice as much of the 50% solution as the 30% solution, how many liters of each solution should be used?

YOU DO #4 : Which of the following would be your variables?

A. x = amount 40% solutionB. y = # of litersC. x = # of batchesD. y = amount of 60% solution

You Do #4: Which of the following equations would you use to solve the problem?

A. x + y = 8B. .4x + .6y = 4.4C. x + y = 55D. .4x + .6y = .55

What is the solution to YOU DO #4

A. 6 liters of the 40% solution and 4 liters of the 60% solution

B. 21.25 liters of the 40% solution and 13.25 liters of the 60% solution

C. 2 liters of the 60% solution and 6 liters of the 40% solution

D. 2 liters of the 40% solution and 6 liters of the 60% solution

Example 5

The sum of two numbers is -11. Twice the first number minus the second is 32. Find the numbers.

YOU DO #5 : Which of the following would be your variables?

A. x = first integerB. y = second integerC. x = sum of integersD. y = difference of integers

You Do #5: Which of the following equations would you use to solve the problem?

A. x + y = 12B. x – y = 12C. x + y = 38D. x – y = 38

What is the solution to YOU DO #5

A. The first integer is 13 and the second integer is 25

B. The first integer is 38 and the second integer is 12

C. The first integer is 25 and the second integer is 13

D. The first integer is 26 and the second integer is 14

White board practice!

Carla bought 3 shirts, 4 pairs of pants, and 2 pairs of shoes for a total of $149.79. Beth bought 5 shirts, 3 pairs of pants, and 3 pairs of shoes for $183.19. Kayla bought 6 shirts, 5 pairs of pants, and a pair of shoes for $181.14. Assume that all of the shirts were the same price, all of the pants were the same price, and all of the shoes were the same price. What was the price of each item? Step one: What are your variables?

A. x = # of shirtsB. y = # of pairs of pantsC. z = # of pairs of shoesD. x = price of a shirtE. y = price of a pair of pantsF. z = price of a pair of shoes

Carla bought 3 shirts, 4 pairs of pants, and 2 pairs of shoes for a total of $149.79. Beth bought 5 shirts, 3 pairs of pants, and 3 pairs of shoes for $183.19. Kayla bought 6 shirts, 5 pairs of pants, and a pair of shoes for $181.14. Assume that all of the shirts were the same price, all of the pants were the same price, and all of the shoes were the same price. What was the price of each item? Step two: What are your equations?

A. x + y + z = 149.79B. 3x + 4y + 2z = 149.79C. 5x + 3y + 3z = 183.19D. 3x + 5y + 6z = 149.79E. 14x + 12y + 6z = 181.14F. 6x + 5y + z = 181.14

Carla bought 3 shirts, 4 pairs of pants, and 2 pairs of shoes for a total of $149.79. Beth bought 5 shirts, 3 pairs of pants, and 3 pairs of shoes for $183.19. Kayla bought 6 shirts, 5 pairs of pants, and a pair of shoes for $181.14. Assume that all of the shirts were the same price, all of the pants were the same price, and all of the shoes were the same price. What was the price of each item? Step three: Find the price of each item. (Click in your answer for the price of a pair of shoes)

23.490

Tristen is training for his pilot’s license. Flight instruction cost $105 per hour, and the simulator costs $45 per hour. The school requires students to spend 4 more hours in airplane training than in the simulator. If Tristen can afford to spend $3870 on training, how many hours can he spend training in an airplane and in a simulator? Step one: What are the variables of this problem?

A. x = cost of flying an airplaneB. x = # of hours in airplane. C. y = # of hours in the simulatorD. Y = cost of flying in the simulator

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Tristen is training for his pilot’s license. Flight instruction cost $105 per hour, and the simulator costs $45 per hour. The school requires students to spend 4 more hours in airplane training than in the simulator. If Tristen can afford to spend $3870 on training, how many hours can he spend training in an airplane and in a simulator. Step 2: What equations can be used to solve this problem?

A. x + y = 3870B. 150x + 45y = 3870C. x = y + 4D. y = x + 4

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Tristen is training for his pilot’s license. Flight instruction cost $105 per hour, and the simulator costs $45 per hour. The school requires students to spend 4 more hours in airplane training than in the simulator. If Tristen can afford to spend $3870 on training, how many hours can he spend training in an airplane. Step 3: Solve the problem using any method.

270.0

Valery is preparing an acid solution. She needs 200 milliliters of 48% concentration solution. Valery has 60% and 40% concentration solutions in her lab. How many milliliters of 40% acid solution should be mixed with 60% acid solution to make the required amount 48% acid solution? Step one: Name the variables

A. x = amount of 60% solutionB. y = amount of 48% solutionC. y = amount of 40% solutionD. x = amount of 200 milliliter

solution

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Valery is preparing an acid solution. She needs 200 milliliters of 48% concentration solution. Valery has 60% and 40% concentration solutions in her lab. How many milliliters of 40% acid solution should be mixed with 60% acid solution to make the required amount 48% acid solution? Step two: Write the equations

A. x + y = .48(200)B. .4x + .6y = .48(200)C. .6x + .4y = .48(200)D. x + y = 200

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Valery is preparing an acid solution. She needs 200 milliliters of 48% concentration solution. Valery has 60% and 40% concentration solutions in her lab. How many milliliters of 40% acid solution should be mixed with 60% acid solution to make the required amount 48% acid solution? Step 3: Solve using matrices.

1200.0

Jambalaya is a Cajun dish made from chicken, sausage, and rice. Simone is making a large pot of jambalaya for a party. Chicken cost $6 per pound, sausage cost $3 per pound, and rice costs $1 per pound. She spends $42 on 13 and a half pounds of food. She buys twice as much rice as sausage. How much sausage will she use in her dish. Step one: Name your variables

A. x – cost of the chickenB. y – cost of the sausageC. z – cost of the riceD. x – amount of chickenE. y – amount of sausageF. z – amount of rice

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Jambalaya is a Cajun dish made from chicken, sausage, and rice. Simone is making a large pot of jambalaya for a party. Chicken cost $6 per pound, sausage cost $3 per pound, and rice costs $1 per pound. She spends $42 on 13 and a half pounds of food. She buys twice as much rice as sausage. How much sausage will she use in her dish. Step two: Make your equations.

A. x + y + z = 42B. x + y + z = 13.5C. z = 2yD. y = 2zE. 6x + 3y + z = 42F. 6x + 3y +z = 13.5

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Jambalaya is a Cajun dish made from chicken, sausage, and rice. Simone is making a large pot of jambalaya for a party. Chicken cost $6 per pound, sausage cost $3 per pound, and rice costs $1 per pound. She spends $42 on 13 and a half pounds of food. She buys twice as much rice as sausage. How much sausage will she use in her dish. Step three: solve.

30.0

Devonte loves the lunch combinations at Rosita’s Mexican Restaurant. Today however, she wants a different combination than the ones listed on the menu.

7.800.0

Lunch Combo Meals: 1.Two Tacos, One Burrito .….$6.55

2.One Enchilada, One Taco, One Burritio……………………………..$7.10

3.Two Enchiladas, Two Tacos………………………….$8.90If Devonte wants to make a combo

of 2 burritos and 1 enchilada, how much should he plan to spend?

The sum of 3 numbers is 20. The second number is 4 times the first, and the sum of the first and third is 8. Find the numbers.

A. 3B. 15C. 12D. 5

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