Unravel the System: Solve for x and y in Fractional Equations

To solve this problem, we'll break it down into clear steps:

• Step 1: Simplify the given equations for easier manipulation.
• Step 2: Use the substitution method to express one variable in terms of the other from one equation.
• Step 3: Insert this expression into the other equation to solve for one variable.
• Step 4: Finally, substitute back to find the second variable.

Now, let's work through each step:

Step 1: Simplify the first equation.
The first equation is: $\frac{x-y}{2} + \frac{-y+x}{3} = -6$
First, let's find a common denominator of 6 for the fractions: $\frac{3(x-y) + 2(-y+x)}{6} = -6$
This simplifies to: $\frac{3x-3y-2y+2x}{6} = -6$
$\frac{5x-5y}{6} = -6$

Step 1 (cont.): Multiply both sides by 6 to get rid of the denominator:
$5x - 5y = -36$
Divide every term by 5 to make it simpler: $x - y = -\frac{36}{5}$ $x = y - \frac{36}{5}$ (Equation 1)

Step 2: Simplify the second equation.
The second equation is: $\frac{-y-x}{5} - (y + x) = 8$ Multiply through by 5 to clear the fraction from the first term: $-y-x - 5(y + x) = 40$ This expands to: $-y-x - 5y - 5x = 40$ Combine like terms: $-6y - 6x = 40$ Divide every term by -6 for simplicity: $y + x = -\frac{40}{6}$ $y + x = -\frac{20}{3}$ (Equation 2)

Step 3: Substitute for $x$ in Equation 2 using Equation 1.
From (Equation 1), $x = y - \frac{36}{5}$. Substitute this into Equation 2: $y + (y - \frac{36}{5}) = -\frac{20}{3}$ This gives: $2y - \frac{36}{5} = -\frac{20}{3}$ Add $\frac{36}{5}$ to both sides to isolate $2y$: $2y = -\frac{20}{3} + \frac{36}{5}$

Convert both terms to a common denominator to add them together. The common denominator of 3 and 5 is 15: $2y = -\frac{100}{15} + \frac{108}{15}$ This simplifies to: $2y = \frac{8}{15}$ Divide both sides by 2 to solve for $y$: $y = \frac{4}{15} \approx 0.266$

Step 4: Solve for $x$ using $y$'s value in Equation 1.
Plug $y = \frac{4}{15}$ back into Equation 1: $x = \frac{4}{15} - \frac{36}{5}$ To add, convert to a common denominator, which can be 15: $x = \frac{4}{15} - \frac{108}{15}$ $x = \frac{4 - 108}{15}$ $x = -\frac{104}{15} \approx -6.933$

Therefore, after solving both variables, the values that satisfy the given system of equations are:
$x \approx -6.933,\ y \approx 0.266$.