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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 Oakdale are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold $35$ turkey sandwiches and $33$ chef salads, for a total of $\$134$ . During the late lunch, $65$ turkey sandwiches and $70$ chef salads were sold, for a total of $\$275$ . How much does each item cost? $\newline$ A turkey sandwich costs $\$\_$ , and a chef salad costs $\$\_$ .

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Solve a system of equations using elimination: word problems

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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 Oakdale are a deluxe turkey sandwich and a chef salad. During early lunch, the cafeteria sold

35

turkey sandwiches and

33

chef salads, for a total of

\$134

. During the late lunch,

65

turkey sandwiches and

70

chef salads were sold, for a total of

\$275

. How much does each item cost?

\newline

A turkey sandwich costs

\$\_

, and a chef salad costs

\$\_

.

Define variables, set equations: Step $1$ : Define the variables and set up the equations based on the given information. $\newline$ Let $x$ be the cost of a turkey sandwich and $y$ be the cost of a chef salad. $\newline$ From early lunch sales: $35x + 33y = 134$ $\newline$ From late lunch sales: $65x + 70y = 275$
Use elimination method: Step $2$ : Use elimination to solve the system of equations. Choose to eliminate $x$ by making the coefficients of $x$ equal. $\newline$ Multiply the first equation by $65$ and the second equation by $35$ : $\newline$ $(65)(35x + 33y) = 65(134)$ $\newline$ $(35)(65x + 70y) = 35(275)$
Simplify equations: Step $3$ : Simplify the equations from Step $2$ . $\newline$ $2275x + 2145y = 8710$ $\newline$ $2275x + 2450y = 9625$
Eliminate x, solve for y: Step $4$ : Subtract the first new equation from the second to eliminate x. $\newline$ $2275x + 2450y - (2275x + 2145y) = 9625 - 8710$ $\newline$ $305y = 915$
Solve for y: Step $5$ : Solve for y. $\newline$ y = $\frac{915}{305}$ $\newline$ y = $3$
Substitute $y$ to solve $x$ : Step $6$ : Substitute $y = 3$ back into the first original equation to solve for $x$ . $35x + 33(3) = 134$ $35x + 99 = 134$ $35x = 35$ $x = 1$

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