A deli has two platters of sandwiches. The first platter costs ​$4444 and you get 22 hamham sandwiches and 55 turkeyturkey sandwiches. The other platter costs ​$5353 and you get 55 hamham sandwiches and 33 turkeyturkey sandwiches. Let x represent the cost of each hamham sandwich and y represent the cost of each turkeyturkey sandwich. What is the system of linear equations for the given​ scenario? What is the cost of each​ sandwich?

Answer 1

To create a system of linear equations based on the information given about the sandwich platters, we can set up two equations using the variables \(x\) and \(y\), where \(x\) represents the cost of each ham sandwich and \(y\) represents the cost of each turkey sandwich.

From the first platter:

• Cost: $4444
• Number of ham sandwiches: 22
• Number of turkey sandwiches: 55

This gives us the first equation: \[ 22x + 55y = 4444 \quad \text{(1)} \]

From the second platter:

• Cost: $5353
• Number of ham sandwiches: 55
• Number of turkey sandwiches: 33

This gives us the second equation: \[ 55x + 33y = 5353 \quad \text{(2)} \]

Now we have the system of linear equations: \[ \begin{align*}

1. & \quad 22x + 55y = 4444 \
2. & \quad 55x + 33y = 5353 \end{align*} \]

Next, let's solve this system of equations. We can start by simplifying both equations.

Equation 1 can be simplified by dividing by 11: \[ 2x + 5y = 404 \quad \text{(1')} \]

Equation 2 can be simplified by dividing by 11: \[ 5x + 3y = 487 \quad \text{(2')} \]

Now we have the simplified system: \[ \begin{align*} 1’) & \quad 2x + 5y = 404 \ 2’) & \quad 5x + 3y = 487 \end{align*} \]

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

Let's multiply the first equation (1') by 5 and the second equation (2') by 2 to align the coefficients of \(x\):

1'): \[ (2x + 5y) \times 5 \Rightarrow 10x + 25y = 2020 \quad \text{(3)} \]

2'): \[ (5x + 3y) \times 2 \Rightarrow 10x + 6y = 974 \quad \text{(4)} \]

Now we can subtract equation (4) from equation (3): \[ (10x + 25y) - (10x + 6y) = 2020 - 974 \] \[ 19y = 1046 \] \[ y = \frac{1046}{19} = 55 \]

Now that we have \(y\), we substitute it back into one of the equations to find \(x\). Using equation (1'): \[ 2x + 5(55) = 404 \] \[ 2x + 275 = 404 \] \[ 2x = 404 - 275 \] \[ 2x = 129 \] \[ x = \frac{129}{2} = 64.5 \]

Thus, the costs of the sandwiches are:

• Cost of each ham sandwich \(x\) is \(64.50\)
• Cost of each turkey sandwich \(y\) is \(55.00\)

In conclusion:

• Cost of ham sandwich: $64.50
• Cost of turkey sandwich: $55.00