How do you solve the following linear system: #2x-y+3z= 7 , 5x-4y-2z= 3 , 4x+y-2z=-4 #?

1 Answer

Answer:

#x=17/81#
#y=-110/81#
#z=47/27#

Explanation:

Let #2x-y+3z=7# 1st equation
Let #5x-4y-2z=3# 2nd equation
Let #4x+y-2z=-4# 3rd equation

Use 3rd equation so that #y# in terms of #x# and #z#

#y=-4+2z-4x# 3rd equation

Use now 1st and 3rd
#2x-y+3z=7# 1st equation
#2x-(-4+2z-4x)+3z=7#

#2x+4-2z+4x+3z=7#
#2x+4x-2z+3z=7-4#
#6x+z=3# Let this be the 4th equation

Use 2nd and 3rd equations:

#5x-4y-2z=3# 2nd equation

#5x-4(-4+2z-4x)-2z=3#
#5x+16-8z+16x-2z=3#
#5x+16x-8z-2z=3-16#
#21x-10z=-13# Let this be the 5th equation

Solve for #x# and #z# using 4th and 5th equations

From 4th equation: #z=3-6x# insert in 5th equation

#21x-10z=-13# 5th equation

#21x-10(3-6x)=-13#

#21x-30+60x=-13#

#81x=30-13#
#81x=17#
#x=17/81#

Go back to the 4th equation , use #x=17/81#

From 4th equation: #z=3-6x#
#z=3-6*17/81#

#z=47/27#
Go back to the 3rd equation and use #x=17/81# and #z=47/27#

#y=-4+2z-4x# 3rd equation
#y=-4+2*47/27-4*17/81#
#y=-110/81#

The solution set:

#x=17/81#
#y=-110/81#
#z=47/27#

have a nice day from the Philippines!