Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ This morning, Ben processed two catering orders at the sandwich shop where he works. The first order was for $7$ trays of club sandwiches and $5$ trays of vegetarian sandwiches, at a cost of $\$103$ . The second order, which cost $\$53$ , was for $5$ trays of club sandwiches and $1$ tray of vegetarian sandwiches. How much do the trays cost? $\newline$ A tray of club sandwiches costs $\$$ _, and a tray of vegetarian sandwiches costs $\$$ _.

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Solve a system of equations using any method: word problems

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

\newline

This morning, Ben processed two catering orders at the sandwich shop where he works. The first order was for

7

trays of club sandwiches and

5

trays of vegetarian sandwiches, at a cost of

\$103

. The second order, which cost

\$53

, was for

5

trays of club sandwiches and

1

tray of vegetarian sandwiches. How much do the trays cost?

\newline

A tray of club sandwiches costs

\$

_____, and a tray of vegetarian sandwiches costs

\$

_____.

Cost Equations: Let's denote the cost of a tray of club sandwiches as $x$ dollars and the cost of a tray of vegetarian sandwiches as $y$ dollars. The first order's cost equation can be written as: $\newline$ $7x + 5y = 103$
System of Equations: The second order's cost equation can be written as: $5x + 1y = 53$
Elimination Method: We now have a system of equations to solve: $\newline$ $7x + 5y = 103$ $\newline$ $5x + y = 53$ $\newline$ We can use either substitution or elimination to solve this system. Let's use the elimination method.
Solving for x: To eliminate $y$ , we can multiply the second equation by $-5$ and add it to the first equation: $\newline$ $-5(5x + y) = -5(53)$ $\newline$ $-25x - 5y = -265$ $\newline$ Now we add this to the first equation: $\newline$ $7x + 5y = 103$ $\newline$ $-25x - 5y = -265$ $\newline$ ----------------- $\newline$ $-18x = -162$
Substitute x: Solving for x, we divide both sides by $-18$ : $\newline$ $-18x / -18 = -162 / -18$ $\newline$ $x = 9$
Solving for y: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use the second equation: $\newline$ $5x + y = 53$ $\newline$ $5(9) + y = 53$ $\newline$ $45 + y = 53$
Solving for y: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use the second equation: $\newline$ $5x + y = 53$ $\newline$ $5(9) + y = 53$ $\newline$ $45 + y = 53$ Solving for $y$ , we subtract $45$ from both sides: $\newline$ $y = 53 - 45$ $\newline$ $y = 8$