We want to solve
#{: (color(white)(aaa)x + 9y + z = -12), (color(white)(aaaaaa)x + y - 9z = 7) ,(-12x + 4y + z = -4):}}#
We start by putting the system in echelon form using Gaussian elimination
#1)# Add #-1# lots of the 1st equation to the second
#{: (color(white)(aaaaa)x + 9y + z = -12), (color(white)(aaaaaaa)-8y - 10z = 19) ,(color(white)(aa)-12x + 4y + z = -4):}}#
#2)# Add #12# lots of equation 1 to equation three
#{: (color(white)(a)x + 9y + z = -12), (color(white)(aaa)-8y - 10z = 19) ,(112y + 13z= -148):}}#
#3)# Add 14 lots of equation 2 to equation three
#{: (x + 9y + z = -12), (color(white)(aa)-8y - 10z = 19) ,( color(white)(aaaaa)-127z= 118):}}#
We have the system in echelon form, so we now back-substitute.
#z=-118/127#
#y=-1/8(19+10(-118/127))=-1233/1016#
#x=(-12-9(-1233/1016)-(-118/127))=-151/1016#
