That is is 6 equations in 6 variables.

First separate the equations:

(1) 4u+ v= 2y+ 2z

(2) 2u- 2v= 2y- 2z

(3) 2u+ 2v= 4y- z

(4) 2y+ 2z= j+ k

(5) 2y- 2z= j- 2k

(6) 4y- z= j- k

Add 2 times (1) to (2) to get 10u= 6y+ 2z or (7) 5u= 3y+ z

Add (2) and (3) to get (8) 6u= 6y- 3z .

Multiply (7) by 6 to get 30u= 18y+ 6z and multiply (8) by 5 to get 30u= 30y- 3z.

That gives 18y+ 6z= 30y- 3z which reduces to 12y= 9z or 4y= 3z. Replacing y by (3/4) z, 30u= (45/2)z- 3z= (39/2)z so that u= (39/30)z= (13/10)z. Replace u and y by those in (1) and solve for v as multiple of z. You can now replace y in (4), (5), and (6) by (3/4)z to get three equations in z, j, and k. Solve those three equations for the three values, then use that value of z to find y, u, and v.