Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ A baseball coach takes her team out for pizza every time they win a game. Not everyone can come each time, so she orders the pizzas based on how many players are coming. After last week's victory, she bought $3$ small pizzas and $2$ large pizzas, the perfect amount for the $14$ players going out for pizza. This week, there were $12$ players going, so she ordered $2$ small pizzas and $2$ large pizzas, which was also the right amount. How many people does each size of pizza feed? $\newline$ Each small pizza feeds $\_$ people, and each large pizza feeds $\_$ people.

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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

A baseball coach takes her team out for pizza every time they win a game. Not everyone can come each time, so she orders the pizzas based on how many players are coming. After last week's victory, she bought

3

small pizzas and

2

large pizzas, the perfect amount for the

14

players going out for pizza. This week, there were

12

players going, so she ordered

2

small pizzas and

2

large pizzas, which was also the right amount. How many people does each size of pizza feed?

\newline

Each small pizza feeds

\_

people, and each large pizza feeds

\_

people.

Define Variables: Let's denote the number of people a small pizza can feed as $x$ and the number of people a large pizza can feed as $y$ . After last week's victory, the coach bought $3$ small pizzas and $2$ large pizzas for $14$ players. This gives us our first equation: $\newline$ $3x + 2y = 14$
First Equation: This week, there were $12$ players going, and the coach ordered $2$ small pizzas and $2$ large pizzas. This gives us our second equation: $\newline$ $2x + 2y = 12$
Second Equation: We now have a system of equations: $\newline$ $3x + 2y = 14$ $\newline$ $2x + 2y = 12$ $\newline$ We can solve this system by subtracting the second equation from the first to eliminate $y$ . $\newline$ $(3x + 2y) - (2x + 2y) = 14 - 12$ $\newline$ $3x - 2x + 2y - 2y = 2$ $\newline$ $x = 2$
Solve System of Equations: 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: $\newline$ $2(2) + 2y = 12$ $\newline$ $4 + 2y = 12$ $\newline$ $2y = 12 - 4$ $\newline$ $2y = 8$ $\newline$ $y = 4$
Substitute and Solve for $x$ : We have found the values for $x$ and $y$ : $x = 2$ $y = 4$ This means each small pizza feeds $2$ people, and each large pizza feeds $4$ people.