Question 1016590
E1  x+y+z+w=1  
E2  x+0y+3z+2w=5
E3  3x-y+0z+w=-12
E4  0x+2y+5z+3w=7

4 equations, 4 unknowns. We need to combine 3 pairs of equation to eliminate a variable. Let's start with x. Note, I added the missing variables using 0 as a coefficient. 

-3x+-3y+-3z+-3w=-3  E1*-3
3x-y+0z+w=-12            E3

0x-4y-3z-2w=-15   sum1

-3x+0y-9z-6w=-15  E2*-3
3x-y+0z+w=-12

0x-y-9z-5w=-27  sum2

2y+5z+3w=7  E4 x already gone
-4y-3z-2w=-15  sum1
-y-9z-5w=-27  sum2 

4y+10z+6w=14   E4*2
-4y-3z-2w=-15   sum1

0y+7z+4w=-1  sum3

-4y-36z-20w=-108  sum2*4
4y+10z+6w=14   E4*2

0y-26z-14w=-94   sum4

49z+28w=-7  sum3*7
-52z-28w=-188   sum4*2

-3z=-195

z=65   go to sum 4

-26(65)-14w=-94

-1690-14w=-94

-14w=1596

w=-114

x+0y+3z+2w=5  E2 

x+3(65)+2(-114)=5

x+195-228=5

x= 38

Last, E1

x+y+z+w=1 

38+y+65-114=1

y= 12

(38,12,65,-114)