A deli has two platters of sandwiches. The first platter costs ​$3838 and you get 22 turkeyturkey sandwiches and 33 roast beefroast beef sandwiches. The other platter costs ​$3737 and you get 33 turkeyturkey sandwiches and 22 roast beefroast beef sandwiches. Let x represent the cost of each turkeyturkey sandwich and y represent the cost of each roast beefroast beef sandwich. What is the system of linear equations for the given​ scenario? What is the cost of each​ sandwich?

Question

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Part 1
Choose the correct system of linear equations below.
A.
22xplus+33y
equals=
3737
33xplus+22y
equals=
3838
B.
33xplus+22y
equals=
3838
33xplus+22y
equals=
3737
C.
22xplus+33y
equals=
3838
22xplus+33y
equals=
3737
D.
22xplus+33y
equals=
3838
33xplus+22y
equals=
3737

Answer 1

To represent the scenario with a system of linear equations, we can analyze the information given for both platters of sandwiches.

For the first platter:

• The cost is $3838.
• It contains 22 turkey sandwiches and 33 roast beef sandwiches.

This gives us the equation: \[ 22x + 33y = 3838 \]

For the second platter:

• The cost is $3737.
• It contains 33 turkey sandwiches and 22 roast beef sandwiches.

This gives us the equation: \[ 33x + 22y = 3737 \]

Combining these, we get the correct system of linear equations:

1. \( 22x + 33y = 3838 \)
2. \( 33x + 22y = 3737 \)

Thus, the correct answer is: D. \( 22x + 33y = 3838 \)
\( 33x + 22y = 3737 \)

Part 2: Solving the System

Now, let's solve the system of equations to find the values of \( x \) (cost of each turkey sandwich) and \( y \) (cost of each roast beef sandwich).

1. The first equation is: \[ 22x + 33y = 3838 \]

2. The second equation is: \[ 33x + 22y = 3737 \]

To solve this system, we can use the method of substitution or elimination. Here, we'll use elimination.

To eliminate \( y \), we can multiply the first equation by 2 and the second by 3:

1. \( 2(22x + 33y) = 2(3838) \) → \( 44x + 66y = 7676 \)
2. \( 3(33x + 22y) = 3(3737) \) → \( 99x + 66y = 11211 \)

Now, we can subtract the first modified equation from the second: \[ (99x + 66y) - (44x + 66y) = 11211 - 7676 \] This simplifies to: \[ 55x = 3535 \] So: \[ x = \frac{3535}{55} = 64.27 \]

Now that we have \( x \), we can substitute it back into either of the original equations to find \( y \). Let's use the first equation: \[ 22(64.27) + 33y = 3838 \] Calculating \( 22 \times 64.27 \): \[ 1414.94 + 33y = 3838 \] Subtracting \( 1414.94 \) from both sides: \[ 33y = 3838 - 1414.94 = 2423.06 \] Now, divide by 33: \[ y = \frac{2423.06}{33} = 73.36 \]

Summary of Results:

• Cost of each turkey sandwich \( x \): approximately $64.27
• Cost of each roast beef sandwich \( y \): approximately $73.36