To determine the correct system of inequalities that represents the given scenario, let's carefully analyze the conditions provided:
1. Room Capacity Constraint:
- The maximum number of people that can attend the cookout is 200.
- If [tex]\( x \)[/tex] represents the number of adults and [tex]\( y \)[/tex] represents the number of campers, the total number of people is [tex]\( x + y \)[/tex].
- Therefore, the inequality for the room capacity can be written as:
[tex]\[
x + y \leq 200
\][/tex]
2. Budget Constraint:
- The budget available for the cookout is [tex]$750.
- Each adult costs $[/tex]4 and each camper costs $3.
- The total cost for [tex]\( x \)[/tex] adults is [tex]\( 4x \)[/tex], and the total cost for [tex]\( y \)[/tex] campers is [tex]\( 3y \)[/tex].
- Therefore, the inequality for the budget can be written as:
[tex]\[
4x + 3y \leq 750
\][/tex]
Combining these two inequalities, we get the system of inequalities:
[tex]\[
\begin{cases}
x + y \leq 200 \\
4x + 3y \leq 750
\end{cases}
\][/tex]
By comparing this system to the options provided:
1. [tex]\(\begin{array}{l}
x + y \leq 200 \\
4 x + 3 y \leq 750
\end{array}\)[/tex]
2. [tex]\(\begin{array}{l}
0 x + y \leq 750 \\
4 x + 3 y \leq 200
\end{array}\)[/tex]
3. [tex]\(\begin{array}{l}
0 x + y \leq 200 \\
3 x + 4 y \leq 750
\end{array}\)[/tex]
4. [tex]\(\begin{array}{l}
0 x + y \leq 750 \\
3 x + 4 y \leq 200
\end{array}\)[/tex]
The correct system of inequalities is clearly the first option:
[tex]\[
\begin{cases}
x + y \leq 200 \\
4x + 3y \leq 750
\end{cases}
\][/tex]
Therefore, the system of inequalities that represents this real-world scenario is:
[tex]\[
\begin{cases}
x + y \leq 200 \\
4x + 3y \leq 750
\end{cases}
\][/tex]
