How do you solve using gaussian elimination or gauss-jordan elimination, #6x+2y+7z=20#, #-4x+2y+3z=15#, #7x-3y+z=25#?

1 Answer
Jan 26, 2016

Answer:

#x = 325/102 #

#y= -165495/2601#

#z = 470/51#

Explanation:

#6x+2y+7z = 20#
#-4x +2y +3z = 15#
#7x - 3y +z = 25#

Gaussian elimination is a process of solving simultaneous equations in more than two variables by repeatedly adding and/or subtracting the equations.
Observing that there is an element #+2y# in each of the first two equations, if we subtract the middle equation from the first one we get
#10x+4z = 5#

If we add #3*#the second equation to #2*#the third equation

#-12x+6y+9z=45#
#14x-6y+2z=50#

we get #2x+11z = 95#

Using a similar process on these two equations in #x# and #z#
#10x+4z = 5#
#10x+55z = 475#
#51z = 470#

#z = 470/51#

#x = (5-4z)/10 = (255 - 1880)/510 =1625/510 = 325/102 #

Select any one of the original equations to solve for #y#

#2y = 20 -6*(325/102) - 7*(470/51)#
#2y = (20*102*51 - 6*325*51 - 7*102*470)/(102*51)#

#y = (104040 - 99450 -335580)/5202#
#y= - 330990/5202 = -165495/2601#