Determine the value of x:
7.
$$\begin{array}{l}4x-y=4\\ 12x+2y=52\end{array}$$

Step 1 ：This is a system of linear equations. We can solve it using substitution or elimination method. I will use the elimination method.

Step 2 ：First, I will multiply the first equation by 2 and the second equation by 1 to make the coefficients of y in both equations the same.

Step 3 ：Then, I will subtract the second equation from the first to eliminate y and solve for x.

Step 4 ：After finding the value of x, I will substitute it into the first equation to find the value of y.

Step 5 ：The solution to the system of equations is $x=3$ and $y=8$. This means that the value of x that satisfies both equations is 3.

Step 6 ：Final Answer: The value of x that satisfies the system of equations is $\boxed{{\displaystyle 3}}$.