Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Today's cafeteria specials at a high school in Cedarburg are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold $44$ turkey sandwiches and $67$ chef salads, for a total of $\$400$ . During the late lunch, $28$ turkey sandwiches and $40$ chef salads were sold, for a total of $\$244$ . How much does each item cost? $\newline$ A turkey sandwich costs $\$$ _, and a chef salad costs $\$$ _.

Full solution

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

\newline

Today's cafeteria specials at a high school in Cedarburg are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold

44

turkey sandwiches and

67

chef salads, for a total of

\$400

. During the late lunch,

28

turkey sandwiches and

40

chef salads were sold, for a total of

\$244

. How much does each item cost?

\newline

A turkey sandwich costs

\$

_____, and a chef salad costs

\$

_____.

Set up equations: Step $1$ : Set up the system of equations based on the given data. $\newline$ Let $x$ be the cost of a turkey sandwich and $y$ be the cost of a chef salad. $\newline$ From early lunch sales: $44x + 67y = 400$ $\newline$ From late lunch sales: $28x + 40y = 244$
Use elimination method: Step $2$ : Use elimination to solve the system of equations. $\newline$ Multiply the first equation by $28$ and the second by $44$ to align the coefficients for $x$ : $\newline$ $(44x + 67y) \times 28 = 400 \times 28$ $\newline$ $(28x + 40y) \times 44 = 244 \times 44$ $\newline$ $1232x + 1876y = 11200$ $\newline$ $1232x + 1760y = 10736$
Solve for y: Step $3$ : Subtract the second equation from the first to eliminate $x$ : $1876y - 1760y = 11200 - 10736$ $116y = 464$ $y = \frac{464}{116}$ $y = 4$
Substitute to find x: Step $4$ : Substitute $y = 4$ back into one of the original equations to find $x$ : $44x + 67(4) = 400$ $44x + 268 = 400$ $44x = 400 - 268$ $44x = 132$ $x = \frac{132}{44}$ $x = 3$