Let's determine the correct system of equations that represents the relationships between the ages of Bethany, Lauren, Amanda, and David based on the given conditions.
1. Lauren is 13 years older than Bethany:
- If [tex]\( x \)[/tex] is Bethany's age, then Lauren's age is [tex]\( x + 13 \)[/tex].
2. David is 11 years older than Amanda:
- If [tex]\( y \)[/tex] is Amanda's age, then David's age is [tex]\( y + 11 \)[/tex].
3. The product of Bethany's and Lauren's ages is twice Amanda's age:
- This means [tex]\( x \cdot (x + 13) = 2y \)[/tex].
- Simplifying this, we have [tex]\( x^2 + 13x = 2y \)[/tex], or [tex]\( y = \frac{1}{2}(x^2 + 13x) \)[/tex].
4. If 20 years are subtracted from both Bethany's and Lauren's ages, the product is equal to David's age:
- This means [tex]\( (x - 20) \cdot (x + 13 - 20) = y + 11 \)[/tex].
- Simplifying inside the parentheses, we get [tex]\( (x - 20)(x - 7) = y + 11 \)[/tex].
- Expanding this, [tex]\( x^2 - 7x - 20x + 140 = y + 11 \)[/tex], or [tex]\( x^2 - 27x + 140 = y + 11 \)[/tex].
- Therefore, [tex]\( y = x^2 - 27x + 129 \)[/tex].
Combining these equations, we get:
[tex]\[
\left\{
\begin{array}{c}
y = \frac{1}{2} x^2 + \frac{13}{2} x \\
y = x^2 - 27x + 129
\end{array}
\right.
\][/tex]
Comparing this system with the given options, we recognize this matches option C:
[tex]\[
C. \left\{
\begin{array}{l}
y = \frac{1}{2} x^2 + \frac{13}{2} x \\
y = x^2 - 27x + 129
\end{array}
\right.
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
