The system of linear equations that you wrote down can be succinctly expressed by arranging the numbers $A_{11}, \dotsc, A_{44}$ in a matrix $A$ and the numbers $B_1, \dotsc,B_4$ and $C_1, \dotsc,C_4$ in vectors $b$ and $c$, respectively. Then the system can be written $b=Ac$, or, more conventionally, $Ac=b$. If you don't understand this notation, you might want to take a look at matrix multiplication.
But this is just a convenient way of collecting all the values that belong together in approriately structured objects -- you don't necessarily have to understand any of that in order to solve the system. The standard algorithm to solve systems of linear equations is Gaussian elimination. That's what you should use if you want to solve a system only once. If you need to solve several systems with the same matrix $A$ (i.e. the same values $A_{11},\dotsc,A_{44}$) but different vectors $b$ (i.e. different values $B_1,\dotsc,B_4$), there are more efficient ways of doing that. Responding to your comment: It's not that other ways of solving systems of linear equations are "better" in general; it depends on what you want to do. If you want to solve several systems with the same matrix $A$, then it makes sense to do as much preprocessing as possible on $A$ before bringing the individual $b$s into play; whereas if you have only one $b$ to solve for, it's easier and faster to just plug it in right from the start.
If you want to learn more about other methods, there are some pointers in this section.