There are three equations with three variables.
Make #y# the subject in all three equations:
#y= -x-z -1" "#..... equation 1
#y = -3x-4z+8" "# ... equation 2
#y= -x+7z-9" "#...equation 3
By equating the equations in pairs we can form two equations with the variables #x and z# and solve them simultaneously
Using equations 1 and 2: #" "y=y#
#" "-x-z-1 = -3x-4z+8#
#3x-x+4z-z=8+1" "larr# re-arrange
#2x+3z=9" "# equation A
Using equations 3 and 2 #" "y=y#
#" "-x+7z-9=-3x-4z+8" "larr# re-arrange
#3x-x+7z+4z=8+9#
#2x+11z=17" "# equation B
Now solve A and B for #x and z#
#" "2x+11z = 17color(white)(mmmmmmmmmmm)A#
#" "2x+3z =9color(white)(mmmmmmmmmmmm)B#
#A-B:" "8z = 8#
#color(white)(mmmmmm)z =1#
#2x +3(1)=9#
#2x +3=9#
#2x =6#
#x=3#
Now find #y# from equation 1
#y= -x-z -1#
#y = -(3)-(1)-1#
#y =-5#
Check with equation 2
#y =-3x-4z+8#
#y = -3(3)-4(1)+8#
#y=-9-4+8#
#y=-5#