To determine the correct system of equations for the problem, let's break down each component based on their relationships and the conditions given.
### Given Relationships:
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 equal to twice Amanda's age:
- The equation for this is: [tex]\( x(x + 13) = 2y \)[/tex].
4. If they subtract 20 years from both Bethany's and Lauren's age, the product is equal to David's age:
- The adjusted ages for Bethany and Lauren are [tex]$x - 20$[/tex] and [tex]$(x + 13) - 20$[/tex] respectively.
- Simplifying, Lauren's adjusted age is [tex]$x - 7$[/tex].
- Hence, the product of the adjusted ages should equal David's age ([tex]$y + 11$[/tex]):
[tex]\[ (x - 20)(x - 7) = y + 11 \][/tex]
### Developing the Equations:
From the conditions:
1. [tex]\( x(x + 13) = 2y \)[/tex] simplifies to:
[tex]\[ x^2 + 13x = 2y \][/tex]
2. [tex]\((x - 20)(x - 7) = y + 11\)[/tex] simplifies to:
[tex]\[ x^2 - 27x + 140 = y + 11 \][/tex]
Thus,
[tex]\[ x^2 - 27x + 129 = y \][/tex]
### Formulated System:
From above, we have:
1. [tex]\( y = \frac{1}{2}(x^2 + 13x) \)[/tex]
2. [tex]\( y = x^2 - 27x + 129 \)[/tex]
### Checking Options:
Option A: [tex]\(\left\{\begin{array}{l}y = x^2 + \frac{13}{2} x \\ y = x^2 - 27 x + 129\end{array}\right.\)[/tex]
- For [tex]\( y = \frac{1}{2}(x^2 + 13x) \)[/tex], incorrect due to coefficient discrepancy.
Option B: [tex]\(\left\{\begin{array}{l}y = \frac{1}{2} x^2 + \frac{13}{2} x \\ y = 13 x^2 + 27 x + 7 \end{array}\right.\)[/tex]
- Equation 2 incorrect based on derived.
Option C: [tex]\(\left\{\begin{array}{l}y = \frac{1}{2} x^2 + \frac{18}{2} x \\ y = x^2 - 27 x + 129\end{array}\right.\)[/tex]
- For [tex]\( y = \frac{1}{2}(x^2 + 13x) \)[/tex], inaccurate due to extra coefficient 18.
Option D: [tex]\(\left\{\begin{array}{l}y = x^2 + \frac{13}{2} x - 260 \\ y = x^2 - 27 x + 151\end{array}\right.\)[/tex]
- Both incorrect based on derived equations.
Thus, the correct set of equations as concluded is:
[tex]\(\boxed{A. \left\{\begin{array}{l}y = x^2 + \frac{13}{2} x \\ y = x^2 - 27 x + 129\end{array}\right.}\)[/tex]
