How do you solve using gaussian elimination or gauss-jordan elimination, #3w-x=2y + z -4#, #9x-y + z =10#, #4w+3y-z=7#, #12x + 17=2y-z+6#?

1 Answer
Feb 18, 2016

Answer:

#w = -183/735#

#x = -111/49#

#y = 27297/1470#

#z = 71967/1470#

Explanation:

Gaussian elimination involves the gradual elimination of variables from the set of equations until values can be found for each variable.
#3w-x=2y+z-4#
#9x-y+z=10#
#4w+3y-z=7#
#12x+17=2y-z+6#

First reorder the terms in all equations so that they all have the same order of variables.
#3w-x-2y-z=-4#
#9x-y+z=10#
#4w+3y-z=7#
#12x-2y+z=-11#

Adding the first two equations together gives
#3w+8x-3y=6#
Adding the second pair of equations gives
#4w+12x+y =-4#
Adding the middle pair of equations gives
#4w+9x+2y =17#

In a similar process, add three times the middle equation to the first equation
#15w+44x = -6#
Take twice the middle equation from the third equation
#-4w-15x=9#

Now add four times the first equation to fifteen times the second equation
#176x - 225x = -24 +135#
#-49x = 111#

#x = -111/49#

Substitute this back into one of the final pair of equations to get #w#
#15w = -6 -(-111/49) = -183/49#
#w = -183/735#

#2y = 17 -4(-183/735) -9(-111/49) = (12495 -183+14985)/735#
#y = 27297/1470#

#z = 10-9x+y = 10-9(-111/49) +27297/1470#
#=(14700+29970+27297)/1470 = 71967/1470#