To solve this problem, we need to set up a system of inequalities that reflects the given conditions:
1. The first condition is related to the seating capacity, which is limited to 350 people. Therefore, the total number of attendees donating at any level (whether [tex]$100$[/tex] or [tex]$150$[/tex]) must be less than or equal to this capacity. This can be represented mathematically as:
[tex]\[
x + y \leq 350
\][/tex]
where [tex]\(x\)[/tex] is the number of attendees donating [tex]$100, and \(y\) is the number of attendees donating $[/tex]150.
2. The second condition is the goal to raise at least [tex]$50,000. The total amount raised from donations can be described by combining the contributions from both levels of donations. Specifically, attendees donating $[/tex]100 contribute [tex]\(100x\)[/tex] and those donating $150 contribute [tex]\(150y\)[/tex]. Therefore, the inequality representing the goal can be written as:
[tex]\[
100x + 150y \geq 50,000
\][/tex]
So, the system of inequalities that meets these conditions is:
[tex]\[
\begin{aligned}
x + y & \leq 350 \\
100x + 150y & \geq 50,000
\end{aligned}
\][/tex]
Among the given options, the one that matches this system is:
A. [tex]\[
\begin{aligned}
x + y & \leq 350 \\
100x + 150y & \geq 50,000
\end{aligned}
\][/tex]
Therefore, the correct option is:
[tex]\[
1
\][/tex]
