#x+y+z=-1, 3x+y+4z=8,-x-y+7z=9#?

2 Answers
Nov 9, 2017

Answer:

#x=3#
#y=-5#
#z=1#

Explanation:

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#

Nov 9, 2017

Answer:

#x=3#, #y=-5# and #z=1#

Explanation:

#x+y+z=-1#, #3x+y+4z=8# an #-x-y+7z=9#

From first equation, #z=-x-y-1#

Plug #z# into second an third ones;

#3x+y+4*(-x-y-1)=8#

#3x+y-4x-4y-4=8#

#-x-3y=12#

#-x-y+7*(-x-y-1)=9#

#-x-y-7x-7y-7=9#

#-8x-8y=16#

#-8*(x+y)=16# or #x+y=-2#

From second one, #x=-3y-12#

Plug #x# into third one;

#(-3y-12)+y=-2#

#-2y-12=-2#

#-2y=10#, so #y=-5#

Hence #x=-3y-12=(-3)*(-5)-12=3#

Thus, #z=-x-y-1=-3-(-5)-1=1#