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#)
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# )
Consider the system:
What equation must
1 Answer
Explanation:
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
#((0, 12, 12, |, a + c), (0, 28, 28, |, b + 2c), (2, 9, 8, |, c))#
Multiplying row 1 by
#((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#