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