How do you solve using gaussian elimination or gauss-jordan elimination, #x+3y-6z=7#, #2x-y+2z=0#, #x+y+2z=-1#?

1 Answer
Jan 18, 2016

Answer:

#x=1#
#y=0#
#z=-1#

Explanation:

Initial Augmented Matrix:
#[ ( 1.00, 3.00, -6.00, 7.00), ( 2.00, -1.00, 2.00, 0.00), ( 1.00, 1.00, 2.00, -1.00) ]#

Pivot Action #n#
pivot row = n; pivot column = n; pivot entry augmented matrix entry at (n,n)

1. convert pivot n row so pivot entry = 1
2. adjust non-pivot rows so entries in pivot column = 0

Pivot #color(black)(1)#
Pivot Row 1 reduced by dividing all entries by 1.00 so pivot entry = 1
#[ ( 1.00, 3.00, -6.00, 7.00), ( 2.00, -1.00, 2.00, 0.00), ( 1.00, 1.00, 2.00, -1.00) ]#

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 1 from each non-pivot row

#[ ( 1.00, 3.00, -6.00, 7.00), ( 0.00, -7.00, 14.00, -14.00), ( 0.00, -2.00, 8.00, -8.00) ]#

Pivot #color(black)(2)#
Pivot Row 2 reduced by dividing all entries by -7.00 so pivot entry = 1
#[ ( 1.00, 3.00, -6.00, 7.00), ( 0.00, 1.00, -2.00, 2.00), ( 0.00, -2.00, 8.00, -8.00) ]#

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 2 from each non-pivot row

#[ ( 1.00, 0.00, 0.00, 1.00), ( 0.00, 1.00, -2.00, 2.00), ( 0.00, 0.00, 4.00, -4.00) ]#

Pivot #color(black)(3)#
Pivot Row 3 reduced by dividing all entries by 4.00 so pivot entry = 1
#[ ( 1.00, 0.00, 0.00, 1.00), ( 0.00, 1.00, -2.00, 2.00), ( 0.00, 0.00, 1.00, -1.00) ]#

Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 3 from each non-pivot row

#[ ( 1.00, 0.00, 0.00, 1.00), ( 0.00, 1.00, 0.00, 0.00), ( 0.00, 0.00, 1.00, -1.00) ]#