How do you solve using gaussian elimination or gauss-jordan elimination, #2x + 5y - 2z = 14#, #5x -6y + 2z = 0#, #4x - y + 3z = -7#?

2 Answers
kubik98
Jan 6, 2018

#P={(2,0,-5)}#

Explanation:

#([2,5,-2,|,14],[5,-6,2,|,0],[4,-1,3,|,-7])~~([10,25,-10,|,70],[10,-12,4,|,0],[0,-11,7,|,-35])#
#R_3=R_3-2xxR_1#
#R_2=R_2xx2#
#R_1=R_1xx5#
#R_2=R_2-R_1#

#([10,25,-10,|,70],[0,-37,14,|,-70],[0,-11,7,|,-35])~~([10,25,-10,|,70],[0,-407,154,|,-770],[0,-407,259,|,-1295])#
#R_2=R_2xx11#
#R_3=R_3xx37#
#R_3=R_3-R_2#

#([10,25,-10,|,70],[0,-407,154,|,-770],[0,0,105,|,-525])~~([10,25,-10,|,70],[0,-37,14,|,-70],[0,0,1,|,-5])#
#R_3=R_3xx1/105#
#R_2=R_2xx1/11#
#R_2=R_2-14xxR_3#
#R_1=R_1+10xxR_3#

#([10,25,0,|,20],[0,-37,0,|,0],[0,0,1,|,-5])#
#R_2=R_2xx-1/37#
#R_1=R_1-25xxR_2#
#R_1=R_1xx1/10#

#([1,0,0,|,2],[0,1,0,|,0],[0,0,1,|,-5])#

#P={(2,0,-5)}#

Cem Sentin
Jan 7, 2018

#x=2#, #y=0# and #z=-5#

Explanation:

From third equation, #y=4x+3z+7#

Hence,

#2x+5*(4x+3z+7)-2z=14# or #22x+13z=-21# #(1)# and,

#5x-6*(4x+3z+7)+2z=0# or #-19x-16z=42# #(2)#

From #(1)#, #z=(-22x-21)/13#

Consequently,

#-19x-16*(-22x-21)/13=42#

#(352x+336)/13-19x=42#

#(352x+336)/13=19x+42#

#352x+336=13*(19x+42)#

#352x+336=247x+546#

#105x=210#, so #x=2#

Thus,

#z=((-22)*2-21)/13=-5# and,

#y=4*2+3*(-5)+7=0#

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