Solve the set of linear equations for variable x: $\begin{array}{l}4 x-y=8 \\ 5 x+3 y=27\end{array}$

Step 1 ：The given system of equations is a set of linear equations. We can solve this system using various methods such as substitution, elimination or matrix method. Here, I will use the elimination method.

Step 2 ：In the elimination method, we either add or subtract the equations in order to eliminate one of the variables. In this case, I will multiply the first equation by 3 and the second equation by 1, then add the two equations together. This will eliminate the variable y, and we can solve for x.

Step 3 ：After finding the value of x, we can substitute it into one of the original equations to find the value of y.

Step 4 ：The solution to the system of equations is \(x = 3\) and \(y = 4\). This means that these values satisfy both equations.

Step 5 ：Final Answer: The solution to the system of equations is \(\boxed{x = 3}\) and \(\boxed{y = 4}\).