Detailed Equation Analysis: Decipher the System -2x+3y=4, x-4y=8

To solve this problem, we'll follow these specific steps:

• First, look at our system of equations:
• Equation 1: $-2x + 3y = 4$
• Equation 2: $x - 4y = 8$
• We choose to use the elimination method to remove one variable from the equations. We'll aim to eliminate $x$.
• To achieve this, multiply the second equation by 2 so that we can align the coefficients of $x$ in both equations:
• New Equation 2: $2x - 8y = 16$
• Now, add the transformed second equation to Equation 1 to cancel out $x$:
$(-2x + 3y) + (2x - 8y) = 4 + 16$
• This simplifies to:
$-5y = 20$
• Solve for $y$:
$y = -4$
• With $y$ known, substitute back into the second original equation to determine $x$:
$x - 4(-4) = 8$
• Simplify and solve for $x$:
$x + 16 = 8 \quad \Rightarrow \quad x = 8 - 16 \quad \Rightarrow \quad x = -8$

We have now found the solution for the system of equations. The values are $x = -8$ and $y = -4$.

Thus, the correct answer choice is $x = -8, y = -4$.