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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

This morning, Kenny processed two catering orders at the sandwich shop where he works. The first order was for 4 trays of club sandwiches and 4 trays of vegetarian sandwiches, at a cost of $72. The second order, which cost $42, was for 1 tray of club sandwiches and 4 trays of vegetarian sandwiches. How much do the trays cost?

A tray of club sandwiches costs $
, and a tray of vegetarian sandwiches costs $
.

11 answers

Let \( x \) be the cost of a tray of club sandwiches, and \( y \) be the cost of a tray of vegetarian sandwiches.

We can set up the following system of equations based on the information given:

  1. For the first order (4 trays of club sandwiches and 4 trays of vegetarian sandwiches with a total cost of $72): \[ 4x + 4y = 72 \]

  2. For the second order (1 tray of club sandwiches and 4 trays of vegetarian sandwiches with a total cost of $42): \[ 1x + 4y = 42 \]

Now, we can simplify the first equation by dividing everything by 4: \[ x + y = 18 \quad (1) \]

Now we have the simplified first equation (1) and the second equation:

  1. \( x + y = 18 \)
  2. \( x + 4y = 42 \)

Next, we can solve the system using substitution or elimination. Here, we will use substitution. From equation (1), we can express \( x \) in terms of \( y \): \[ x = 18 - y \quad (2) \]

Now substitute (2) into the second equation: \[ (18 - y) + 4y = 42 \]

Combine like terms: \[ 18 + 3y = 42 \]

Now, isolate \( y \): \[ 3y = 42 - 18 \] \[ 3y = 24 \] \[ y = 8 \]

Now, substitute \( y \) back into equation (2) to find \( x \): \[ x = 18 - 8 \] \[ x = 10 \]

Thus, the cost of a tray of club sandwiches is \( x = 10 \) dollars, and the cost of a tray of vegetarian sandwiches is \( y = 8 \) dollars.

A tray of club sandwiches costs $10, and a tray of vegetarian sandwiches costs $8.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

The box office at a theater is selling tickets for a series of rock concerts. So far, they have sold 85 balcony tickets and 67 general admission floor tickets for Friday's show, for a total of $5,142 in receipts. For Saturday's show, 88 balcony tickets and 67 general admission floor tickets have been sold, equaling $5,262 in receipts. How much does each ticket cost?

A balcony seat ticket costs $
, and a general admission floor ticket costs $
.

Let \( x \) be the cost of a balcony ticket and \( y \) be the cost of a general admission floor ticket.

We can set up the following system of equations based on the information provided:

  1. For Friday's show (85 balcony tickets and 67 general admission floor tickets with total receipts of $5,142): \[ 85x + 67y = 5142 \quad (1) \]

  2. For Saturday's show (88 balcony tickets and 67 general admission floor tickets with total receipts of $5,262): \[ 88x + 67y = 5262 \quad (2) \]

Now we can solve this system using the elimination method. We can eliminate \( y \) by subtracting equation (1) from equation (2):

Subtract (1) from (2): \[ (88x + 67y) - (85x + 67y) = 5262 - 5142 \]

This simplifies to: \[ 3x = 120 \]

Now, solve for \( x \): \[ x = \frac{120}{3} = 40 \]

Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use equation (1): \[ 85(40) + 67y = 5142 \] \[ 3400 + 67y = 5142 \] \[ 67y = 5142 - 3400 \] \[ 67y = 1742 \] \[ y = \frac{1742}{67} = 26 \]

Thus, the cost of a balcony ticket is \( x = 40 \) dollars, and the cost of a general admission floor ticket is \( y = 26 \) dollars.

A balcony seat ticket costs $40, and a general admission floor ticket costs $26.

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Myra is going to ship some gifts to family members, and she is considering two shipping companies. The first shipping company charges a fee of $12 to ship a medium box, plus an additional $5 per kilogram. A second shipping company charges $14 for the same size of box, plus an additional $4 per kilogram. At a certain weight, the two shipping methods will cost the same amount. How much will it cost? What is that weight?

The two shipping methods both cost $
at a weight of
kilograms.

Let \( x \) represent the weight of the package in kilograms. We need to set up equations for the cost of shipping with each company.

  1. Cost with the first shipping company: The cost is given by: \[ C_1 = 12 + 5x \]

  2. Cost with the second shipping company: The cost is given by: \[ C_2 = 14 + 4x \]

We want to find the weight \( x \) at which these two costs are equal, so we set the two equations equal to one another: \[ 12 + 5x = 14 + 4x \]

Now, isolate \( x \):

  1. Subtract \( 4x \) from both sides: \[ 12 + 5x - 4x = 14 \] \[ 12 + x = 14 \]

  2. Subtract 12 from both sides: \[ x = 14 - 12 \] \[ x = 2 \]

Now that we have the weight, we can calculate the cost at that weight using either cost equation. We will use the first shipping company’s cost equation: \[ C_1 = 12 + 5x = 12 + 5(2) = 12 + 10 = 22 \]

Thus, the two shipping methods both cost $22 at a weight of 2 kilograms.

To summarize:

  • The two shipping methods both cost $22 at a weight of 2 kilograms.
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Bridget's Bakery recently spent a total of $301 on new equipment, and their average hourly operating costs are $18. Their average hourly receipts are $19. The bakery will soon make back the amount it invested in equipment. What would the total expenses and receipts both equal? How many hours will that take?

The bakery's expenses and receipts will both total $
after
hours.

Let's set up a system of equations to analyze the situation at Bridget's Bakery.

  1. Let \( h \) be the number of hours the bakery operates.

Equations:

  • Bridget's Bakery spent a total of $301 on new equipment.

  • Their average hourly operating costs are $18, so the total expenses can be expressed as: \[ \text{Total Expenses} = 301 + 18h \quad (1) \]

  • Their average hourly receipts are $19, so the total receipts can be expressed as: \[ \text{Total Receipts} = 19h \quad (2) \]

Setting Up the Condition:

We want to find when the total expenses equal the total receipts: \[ 301 + 18h = 19h \]

Solving for \( h \):

Rearranging the equation, we can isolate \( h \):

  1. Subtract \( 18h \) from both sides: \[ 301 = 19h - 18h \] \[ 301 = h \]

Finding Total Expenses and Receipts:

Now that we have \( h = 301 \), we can substitute this back into either equation (1) or (2) to find the total expenses and total receipts.

Using equation (2) for total receipts: \[ \text{Total Receipts} = 19h = 19(301) = 5739 \]

Using equation (1) for total expenses: \[ \text{Total Expenses} = 301 + 18h = 301 + 18(301) = 301 + 5418 = 5739 \]

Conclusion:

Thus, the bakery's expenses and receipts will both total $5739 after 301 hours.

To summarize:

  • The bakery's expenses and receipts will both total $5739 after 301 hours.
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Dr. Stanley, a pediatrician, has 1 annual checkup and 3 sick visits scheduled next Tuesday, which will fill a total of 142 minutes on his schedule. Next Wednesday, he has 3 annual checkups and 3 sick visits on the schedule, which should take 246 minutes. How much time is allotted for each type of appointment?

The time allotted is
minutes for an annual checkup and
minutes for a sick visit.

Let \( x \) be the time (in minutes) allotted for an annual checkup, and \( y \) be the time (in minutes) allotted for a sick visit.

We can set up the following system of equations based on the information provided:

  1. For next Tuesday (1 annual checkup and 3 sick visits, taking a total of 142 minutes): \[ x + 3y = 142 \quad (1) \]

  2. For next Wednesday (3 annual checkups and 3 sick visits, taking a total of 246 minutes): \[ 3x + 3y = 246 \quad (2) \]

Now, we can simplify equation (2) by dividing everything by 3: \[ x + y = 82 \quad (3) \]

Now, we have a new system of two equations:

  1. \( x + 3y = 142 \) (Equation 1)
  2. \( x + y = 82 \) (Equation 3)

Next, we can solve the system using the elimination method or substitution. Here, we will use substitution. We can solve equation (3) for \( x \): \[ x = 82 - y \quad (4) \]

Now, substitute equation (4) into equation (1): \[ (82 - y) + 3y = 142 \]

Now combine like terms: \[ 82 - y + 3y = 142 \] \[ 82 + 2y = 142 \]

Next, isolate \( y \): \[ 2y = 142 - 82 \] \[ 2y = 60 \] \[ y = 30 \]

Now substitute \( y \) back into equation (4) to find \( x \): \[ x = 82 - 30 \] \[ x = 52 \]

Thus, the time allotted is 52 minutes for an annual checkup and 30 minutes for a sick visit.

To summarize:

  • The time allotted is 52 minutes for an annual checkup and 30 minutes for a sick visit.
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Grace is using ribbon to create girls' hair barrettes. For a craft fair in Georgetown, she made 12 small barrettes and 14 large barrettes, using a total of 172 meters of ribbon. Then, for another craft fair in Weston, she made 15 small barrettes and 13 large barrettes, which used a total of 179 meters. How many yards of ribbon does Grace use for each?

Grace uses
meters of ribbon on each small barrette and
meters on each large one.

Let \( x \) be the amount of ribbon (in meters) used for each small barrette, and \( y \) be the amount of ribbon (in meters) used for each large barrette.

We can set up the following system of equations based on the information provided:

  1. For the Georgetown craft fair (12 small barrettes and 14 large barrettes using a total of 172 meters of ribbon): \[ 12x + 14y = 172 \quad (1) \]

  2. For the Weston craft fair (15 small barrettes and 13 large barrettes using a total of 179 meters of ribbon): \[ 15x + 13y = 179 \quad (2) \]

Solving the System of Equations

To solve this system of equations, we can use the elimination method. First, let's manipulate both equations to eliminate one of the variables. We can align the coefficients of \( y \) in both equations.

Multiply equation (1) by 13: \[ 13(12x + 14y) = 13(172) \] \[ 156x + 182y = 2236 \quad (3) \]

Multiply equation (2) by 14: \[ 14(15x + 13y) = 14(179) \] \[ 210x + 182y = 2506 \quad (4) \]

Now we have the system:

  1. \( 156x + 182y = 2236 \) (from equation 3)
  2. \( 210x + 182y = 2506 \) (from equation 4)

Now subtract equation (3) from equation (4): \[ (210x + 182y) - (156x + 182y) = 2506 - 2236 \] This simplifies to: \[ 54x = 270 \] Now, solve for \( x \): \[ x = \frac{270}{54} = 5 \]

Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use equation (1): \[ 12(5) + 14y = 172 \] \[ 60 + 14y = 172 \] Subtract 60 from both sides: \[ 14y = 172 - 60 \] \[ 14y = 112 \] Now, solve for \( y \): \[ y = \frac{112}{14} = 8 \]

Thus, Grace uses 5 meters of ribbon on each small barrette and 8 meters on each large one.

Summary:

  • Grace uses 5 meters of ribbon on each small barrette and 8 meters on each large one.