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Consider the following linear system.
x + 2y − 3z = a
2x + 3y + 3z = b
5x + 9y − 6z = c
This system is consistent if a, b and c satisfy the equation
- 7a − b − c = 0
- 3a + b − c = 0
- 3a − b + c = 0
- 7a − b + c = 0
Correct answer: 3a + b − c = 0
Solution
The third equation is a linear combination of the first two: 3*(R1) + 1*(R2) gives (5,9,-6) on the left, so consistency requires c = 3a+b, i.e. 3a+b-c=0 (option 1), not 7a-b-c=0.
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