Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. Today's cafeteria specials at a high school in Middletown are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold 21 turkey sandwiches and 17 chef salads, for a total of 110 During the late lunch 12 turkey sandwiches and 56 chef salads were sold, for a total of 248 How much does each item cost? A turkey sandwich costs and a chef salad costs

To solve this problem, we need to set up a system of equations based on the information given. Let's define the variables first:<br /><br />- Let \( x \) be the cost of a turkey sandwich.<br />- Let \( y \) be the cost of a chef salad.<br /><br />We can then write the following system of equations based on the sales during early and late lunch:<br /><br />1. For early lunch:<br />\[ 21x + 17y = 110 \]<br /><br />2. For late lunch:<br />\[ 12x + 56y = 248 \]<br /><br />Now we will solve this system using the elimination method.<br /><br />First, we need to make the coefficients of one of the variables the same in both equations. We'll eliminate \( x \) by finding a common multiple for the coefficients of \( x \) in both equations. The least common multiple of 21 and 12 is 84.<br /><br />We will multiply the first equation by 4 and the second equation by 7 to make the coefficients of \( x \) equal:<br /><br />\[ 4(21x + 17y) = 4(110) \]<br />\[ 84x + 68y = 440 \]<br /><br />\[ 7(12x + 56y) = 7(248) \]<br />\[ 84x + 392y = 1736 \]<br /><br />Now we subtract the first modified equation from the second modified equation to eliminate \( x \):<br /><br />\[ (84x + 392y) - (84x + 68y) = 1736 - 440 \]<br />\[ 84x + 392y - 84x - 68y = 1296 \]<br />\[ 324y = 1296 \]<br />\[ y = \frac{1296}{324} \]<br />\[ y = 4 \]<br /><br />Now that we have the value of \( y \), we can substitute it back into one of the original equations to solve for \( x \). We'll use the first equation:<br /><br />\[ 21x + 17y = 110 \]<br />\[ 21x + 17(4) = 110 \]<br />\[ 21x + 68 = 110 \]<br />\[ 21x = 110 - 68 \]<br />\[ 21x = 42 \]<br />\[ x = \frac{42}{21} \]<br />\[ x = 2 \]<br /><br />So, the cost of a turkey sandwich is \( \$2 \) and the cost of a chef salad is \( \$4 \).<br /><br />A turkey sandwich costs \( \$2 \) and a chef salad costs \( \$4 \).