We are asked to solve the following system of equations using the substitution method: $5x - \frac{2}{3}y = -47$ (1) $\frac{x}{4} + 5y = \frac{51}{4}$ (2)

1. Problem Description

We are asked to solve the following system of equations using the substitution method:

$5x - \frac{2}{3}y = -47$ (1)

$\frac{x}{4} + 5y = \frac{51}{4}$ (2)

2. Solution Steps

First, we solve equation (2) for $x$.

$\frac{x}{4} + 5y = \frac{51}{4}$

Multiply both sides by 4:

$x + 20y = 51$

$x = 51 - 20y$

Next, we substitute this expression for $x$ into equation (1):

$5(51 - 20y) - \frac{2}{3}y = -47$

$255 - 100y - \frac{2}{3}y = -47$

Multiply by 3 to eliminate the fraction:

$3(255 - 100y - \frac{2}{3}y) = 3(-47)$

$765 - 300y - 2y = -141$

$765 - 302y = -141$

$-302y = -141 - 765$

$-302y = -906$

$y = \frac{-906}{-302}$

$y = 3$

Now, substitute $y=3$ back into the equation $x = 51 - 20y$

$x = 51 - 20(3)$

$x = 51 - 60$

$x = -9$

Therefore, the solution is $x=-9$ and $y=3$.

3. Final Answer

(-9, 3)