To determine the correct system of equations representing the given situation, we need to analyze the conditions provided:
1. Each hot dog sold earns \[tex]$0.50.
2. Each hamburger sold earns \$[/tex]0.75.
3. The total number of hot dogs and hamburgers sold is 230.
4. The total revenue earned from the sales of hot dogs and hamburgers is \$138.50.
5. Let [tex]\(x\)[/tex] represent the number of hot dogs sold and [tex]\(y\)[/tex] represent the number of hamburgers sold.
To set up the equations, we translate the given conditions into mathematical expressions:
First Equation (Revenue Equation):
The revenue generated from selling [tex]\(x\)[/tex] hot dogs and [tex]\(y\)[/tex] hamburgers can be expressed as:
[tex]\[0.50x + 0.75y = 138.50\][/tex]
Second Equation (Total Quantity Equation):
The total number of hot dogs and hamburgers sold is given by:
[tex]\[x + y = 230\][/tex]
Therefore, the system of equations representing this situation is:
[tex]\[ \begin{cases}
0.50x + 0.75y = 138.50 \\
x + y = 230
\end{cases}
\][/tex]
Upon comparing with the options provided, we see that:
- Option A: [tex]\(0.50x + 0.75y = 138.50 \quad\text{and}\quad x + y = 230\)[/tex]
This matches our derived system of equations perfectly.
Hence, the correct answer is:
[tex]\[ \boxed{\text{A}} \][/tex]
