Applications of Systems of Equations

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Applications of Systems of Equations
2
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.

3
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?

4
YOU DO 1 Which of the following would be your
variables?

1. x of 3 point questions
2. y amount of money
3. x total of questions
4. y of 5 point questions

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

1. x y 150
2. x y 46
3. 5x 3y 150
4. 3x 5y 150

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What is the solution to YOU DO 1

1. 5 3pt questions and 3 5pt questions
2. 196 3pt questions and 888 5pt questions
3. 40 3pt questions and 6 5pt questions
4. 6 3pt questions and 40 5pt questions

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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?

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

1. x of coins
2. y amount of money
3. x of dimes
4. y of quarters

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

1. x y 100
2. x y 21.40
3. 10x 25y 21.40
4. .10x .25y 21.40

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What is the solution to YOU DO 2

1. 76 quarters and 24 dimes
2. 24 dimes and 76 quarters
3. 165 dimes and -65 quarters
4. 121 dimes and 15 quarters

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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.

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

1. x perimeter
2. y width
3. x length
4. y dimensions

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

1. x y 42
2. 2x 2y 42
3. x y 4
4. y x 4

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What is the solution to YOU DO 3

1. 12.5 ft long X 8.5 ft wide
2. 23 ft long X 19 ft wide
3. 12.5 ft wide X 8.5 ft long
4. 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?

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

1. x amount 40 solution
2. y of liters
3. x of batches
4. y amount of 60 solution

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

1. x y 8
2. .4x .6y 4.4
3. x y 55
4. .4x .6y .55

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What is the solution to YOU DO 4

1. 6 liters of the 40 solution and 4 liters of the
   60 solution
2. 21.25 liters of the 40 solution and 13.25 liters
   of the 60 solution
3. 2 liters of the 60 solution and 6 liters of the
   40 solution
4. 2 liters of the 40 solution and 6 liters of the
   60 solution

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Example 5

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

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

1. x first integer
2. y second integer
3. x sum of integers
4. y difference of integers

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

1. x y 12
2. x y 12
3. x y 38
4. x y 38

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What is the solution to YOU DO 5

1. The first integer is 13 and the second integer is
   25
2. The first integer is 38 and the second integer is
   12
3. The first integer is 25 and the second integer is
   13
4. The first integer is 26 and the second integer is
   14

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White board practice!
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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?

1. x of shirts
2. y of pairs of pants
3. z of pairs of shoes
4. x price of a shirt
5. y price of a pair of pants
6. z price of a pair of shoes

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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?

1. x y z 149.79
2. 3x 4y 2z 149.79
3. 5x 3y 3z 183.19
4. 3x 5y 6z 149.79
5. 14x 12y 6z 181.14
6. 6x 5y z 181.14

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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)

• 23.49
• 0

27
Tristen is training for his pilots 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?

1. x cost of flying an airplane
2. x of hours in airplane.
3. y of hours in the simulator
4. Y cost of flying in the simulator

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28
Tristen is training for his pilots 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?

1. x y 3870
2. 150x 45y 3870
3. x y 4
4. y x 4

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29
Tristen is training for his pilots 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.

• 27
• 0.0

30
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

1. x amount of 60 solution
2. y amount of 48 solution
3. y amount of 40 solution
4. x amount of 200 milliliter solution

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31
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

1. x y .48(200)
2. .4x .6y .48(200)
3. .6x .4y .48(200)
4. x y 200

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32
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.

• 120
• 0.0

33
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

1. x cost of the chicken
2. y cost of the sausage
3. z cost of the rice
4. x amount of chicken
5. y amount of sausage
6. z amount of rice

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34
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.

1. x y z 42
2. x y z 13.5
3. z 2y
4. y 2z
5. 6x 3y z 42
6. 6x 3y z 13.5

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35
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.

• 3
• 0.0

36
Devonte loves the lunch combinations at Rositas
Mexican Restaurant. Today however, she wants a
different combination than the ones listed on the
menu.

• Lunch Combo Meals
• Two Tacos, One Burrito ..6.55
• One Enchilada, One Taco, One Burritio..
  7.10
• Two Enchiladas,
• Two Tacos.8.90

• 7.80
• 0.0

If Devonte wants to make a combo of 2 burritos
and 1 enchilada, how much should he plan to spend?
37
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.

1. 3
2. 15
3. 12
4. 5

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