A summer camp cookout is planned for the campers and their families. There is room for 200 people. Each adult costs [tex]\$4[/tex], and each camper costs [tex]\$3[/tex]. There is a maximum budget of [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[/tex] is the number of campers.

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

B.
[tex]
\begin{array}{l}
0x + y \leq 750 \\
4x + 3y \leq 200
\end{array}
[/tex]

C.
[tex]
\begin{array}{l}
0x + y \leq 200 \\
3x + 4y \leq 750
\end{array}
[/tex]

D.
[tex]
\begin{array}{l}
0x + y \leq 750 \\
3x + 4y \leq 200
\end{array}
[/tex]



Answer :

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]