z+y=16 (adding (2) and (4)); z=16-y.
x=9+y; z=14+w, so w=z-14=16-y-14=2-y.
Therefore, x=9+y, w=2-y and z=16-y: 3 variables in terms of one.
Add all 4 equations: 2x+2z=37; 2(x+z)=37. Substitute for x and z:
2(9+y+16-y)=50=37 which is clearly false, so the equations are inconsistent. There is no solution.
REVISED QUESTION (see related questions)
I suspect that equation (1) should be x+y=9. Assuming this and that all other equations are correct:
z+y=16 (adding (2) and (4)); z=16-y.
x=9-y; z=14+w, so w=z-14=2-y.
Therefore, x=9-y, w=2-y and z=16-y: 3 variables in terms of one.
Add all 4 equations: 2x+2y+2z=37. Substitute for x and z:
2(9-y+y+16-y)=37; 2(25-y)=37; 25-y=18.5, y=25-18.5=6.5 or 13/2.
So w=-9/2; x=5/2; z=19/2.