Technical Explanation: Break Down Each Fraction in the System of Equations

We begin by examining the given system of equations:

$\frac{3}{x} + \frac{1}{y} = 4$ --- (1)

$\frac{5}{x} - \frac{1}{y} = 4$ --- (2)

Let's eliminate $\frac{1}{y}$ by adding equations (1) and (2):

$\left(\frac{3}{x} + \frac{1}{y}\right) + \left(\frac{5}{x} - \frac{1}{y}\right) = 4 + 4$

$\frac{3}{x} + \frac{5}{x} = 8$

$\frac{8}{x} = 8$

Solving for $x$, we have:

$x = 1$

Now, substitute $x = 1$ back into equation (1):

$\frac{3}{1} + \frac{1}{y} = 4$

$3 + \frac{1}{y} = 4$

$\frac{1}{y} = 1$

Solving for $y$, we obtain:

$y = 1$

Thus, the solution to the system of equations is $x = 1$ and $y = 1$.

The correct answer choice is: $x=1,y=1$.