Use matrix transformations to solve the system $$ \begin{alignat}{15} 6x &{}+{}& 3y &{}+{}& 2z &{}+{}& 3u &{}+{}& 4v &{}={}& 6 & \\ 4x &{}+{}& 2y &{}+{}& z &{}+{}& 2u &{}+{}& 3v &{}={}& 4 & \\ 4x &{}+{}& 2y &{}+{}& 3z &{}+{}& 2u &{}+{}& v &{}={}& 4 & \\ 2x &{}+{}& y &{}+{}& 7z &{}+{}& 3u &{}+{}& 2v &{}={}& 7 \end{alignat} $$
Should I turn the matrix that I get after rewriting coefficients into RREF (row reduced echelon form) to make it as easy as it is possible to solve it?
Once you have the Augmented Matrix in RREF, you can easily check whether the system has solution and then solve starting from the bottom pivot.
You can find many other OP on this topic:
Solve system of linear equations in RREF