Consider the system: #{ (−2x_1 + 3x_2 + 4x_3 = a), (−4x_1 + 10x_2 + 12x_3 = b), (2x_1 + 9x_2 + 8x_3 = c) :}# What equation must #a#, #b# and #c# satisfy for this system to be consistent? (i.e. an equation like #5a − 3b + c = 0#)

Question

Consider the system:
#−2x_1 + 3x_2 + 4x_3 = a#
#−4x_1 + 10x_2 + 12x_3 = b#
#2x_1 + 9x_2 + 8x_3 = c#

What equation must
#a#, #b# and #c# satisfy for this system to be consistent? (i.e. an equation like
#5a − 3b + c = 0#)

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Answer 1 · 2017-09-21T05:57:21

#7a-3b+c = 0#

Given:

#{ (−2x_1 + 3x_2 + 4x_3 = a), (−4x_1 + 10x_2 + 12x_3 = b), (2x_1 + 9x_2 + 8x_3 = c) :}#

We can write this as an augmented matrix:

#((-2, 3, 4, |, a), (-4, 10, 12, |, b), (2, 9, 8, |, c))#

Adding row 3 to row 1 and #2 xx # row 3 to row 2, this becomes:

#((0, 12, 12, |, a + c), (0, 28, 28, |, b + 2c), (2, 9, 8, |, c))#

Multiplying row 1 by #7# and row 2 by #3#, this becomes:

#((0, 84, 84, |, 7a + 7c), (0, 84, 84, |, 3b + 6c), (2, 9, 8, |, c))#

Then subtracting row 2 from row 1, we get:

#((0, 0, 0, |, 7a -3b + c), (0, 84, 84, |, 3b + 6c), (2, 9, 8, |, c))#

So:

#7a-3b+c = 0x_1+0x_2+0x_3 = 0#