Bethany,Lauren,Amanda,and David all meet at Family reunion. They are comparing their ages. They have found that Laura is 13 years older than Metheny and David is 11 years older than Amanda. They also found that the product of Bethany and Laura Sage is equal to twice and Amanda's age.Also, if they subtract 20 years from both Bethany's and Louren's age, the product is equal to David's age.If x represents Bethany's age y represents Amanda's age, then which of the following system of equations can be used to determine the age of all for family members based on their findings?

The second system of equations, 

[tex] \left \{ {{y=\frac{x^2}{2}+\frac{13}{2}x} \atop {y=x^2-27x+129}} [/tex]
is correct.

We know that Bethany's age is x.  Since Laura is 13 years older, her age is x+13.  The product of their ages is equal to twice Amanda's age, and Amanda's age is y.  This gives us:
x(x+13) = 2y

Using the distributive property, we have 
x²+13x=2y

Dividing everything by 2 (to isolate y), we have:
x²/2 + (13/2)x = y

If we take 20 years off of Bethany's age, it is now represented as x-20.  Taking 20 years off of Laura's age would be (x+13-20) or x-7.  The product of their ages now is equal to David's age; David is 11 years older than Amanda, so his age is y+11.  This gives us:

(x-20)(x-7)=y+11

Multiplying the binomials we have:"
x*x - 7*x - 20*x - 20(-7) = y+11
x²-7x-20x--140=y+11
x²-27x+140=y+11

To isolate y, subtract 11 from both sides:
x²-27x+140-11 = y+11-11
x²-27x+129 = y