Solve Two Equations Simultaneously: Using Substitution to Find x and y

To solve this system, we'll first simplify and manipulate the equations to perform substitution. Let's begin:

The first equation is $\frac{2x-3y}{4} + \frac{x-y}{5} = 30$.
To eliminate the fractions, multiply through by 20 (the least common multiple of 4 and 5):

$20 \left( \frac{2x-3y}{4} \right) + 20 \left( \frac{x-y}{5} \right) = 20 \times 30$

Simplifying each term gives:
$5(2x - 3y) + 4(x - y) = 600$

Expanding the terms:

$10x - 15y + 4x - 4y = 600$

Combine like terms:

$14x - 19y = 600$ [Equation (1)]

Next, let's work on the second equation: $\frac{2y + x}{8} - \frac{3y - x}{4} = 12$.
Multiply through by 8 to clear the denominators:

$8 \left( \frac{2y + x}{8} \right) - 8 \left( \frac{3y - x}{4} \right) = 8 \times 12$

Simplifying each term gives:

$2y + x - 2(3y - x) = 96$

Expanding the terms:

$2y + x - 6y + 2x = 96$

Combine like terms for a simplified second equation:

$3x - 4y = 96$ [Equation (2)]

Now let's use substitution:

From Equation (2), solve for $x$:

$3x = 4y + 96$

$x = \frac{4y + 96}{3}$

Substitute this expression for $x$ in Equation (1):

$14\left( \frac{4y + 96}{3} \right) - 19y = 600$

Multiply through by 3 to clear the fraction:

$14(4y + 96) - 57y = 1800$

$56y + 1344 - 57y = 1800$

Combine like terms:

$-y + 1344 = 1800$

Solve for $y$:

$-y = 1800 - 1344 = 456$

$y = -456$

Substitute $y = -456$ back into the expression for $x$:

$x = \frac{4(-456) + 96}{3}$

$x = \frac{-1824 + 96}{3}$

$x = \frac{-1728}{3}$

$x = -576$

Thus, the solution for the system is:

$x = -576, y = -456$

After verifying against the provided answer choices, the solution is consistent with choice 4.

Therefore, the solution to the problem is $x = -576, y = -456$.