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]
