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A summer camp cookout is planned for the campers and their families. There is room for 200 people. Each adult costs [tex]$\$[/tex] 4[tex]$, and each camper costs $[/tex]\[tex]$ 3$[/tex]. There is a maximum budget of [tex]$\$[/tex] 750[tex]$. Write the system of inequalities to represent this real-world scenario, where $[/tex]x[tex]$ is the number of adults and $[/tex]y$ is the number of campers.

A.
[tex]\[
\begin{array}{r}
x + y \leq 200 \\
4x + 3y \leq 750
\end{array}
\][/tex]

B.
[tex]\[
\begin{array}{l}
x + y \leq 750 \\
4x + 3y \leq 200
\end{array}
\][/tex]

C.
[tex]\[
\begin{array}{l}
x + y \leq 200 \\
3x \div 4y \leq 750
\end{array}
\][/tex]

D.
[tex]\[
\begin{array}{l}
x + y \leq 750 \\
3x + 4y \leq 200
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's break down the given real-world scenario into a system of inequalities step by step:

1. Constraints Based on Room Capacity:
- The total number of people (adults and campers) that can be accommodated in the room is 200. Therefore, the sum of the number of adults [tex]\( x \)[/tex] and the number of campers [tex]\( y \)[/tex] should be less than or equal to 200.

This can be represented as:
[tex]\[ x + y \leq 200 \][/tex]

2. Constraints Based on Budget:
- The cost for each adult is [tex]$4, and the cost for each camper is $[/tex]3. The total budget for the event is [tex]$750. Therefore, the overall cost for the number of adults \( x \) and the number of campers \( y \) should be less than or equal to $[/tex]750.

This can be represented as:
[tex]\[ 4x + 3y \leq 750 \][/tex]

Putting these constraints together, the system of inequalities for this scenario can be written as:

[tex]\[ \begin{cases} x + y \leq 200 & \text{(Total number of people)} \\ 4x + 3y \leq 750 & \text{(Total cost within budget)} \end{cases} \][/tex]

So, the correct system of inequalities is:

[tex]\[ \begin{array}{r} x + y \leq 200 \\ 4 x + 3 y \leq 750 \end{array} \][/tex]

The correct option is:
[tex]\[ \begin{array}{r} x + y \leq 200 \\ 4 x + 3 y \leq 750 \end{array} \][/tex]

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