Question 1122204: Find values of a, b, and c (if possible) such that the system of linear equations has a unique solution, no solution, and infinitely many solutions. (If not possible, enter IMPOSSIBLE.)
x + y = 4
y + z = 4
x + z = 4
ax + by + cz = 0
Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website! .
The sub-system consisting of 3 (three) first equations has a unique solution x = y = z = 2.
When you add the fourth equation, it becomes
2a + 2b + 2c = 0, or
a + b + c = 0.
Thus,
a) if a + b + c = 0, then the system of 4 equations has a unique solution.
b) if a + b + c =/= 0, then the system of 4 equations has no solution.
c) it is IMPOSSIBLE to this system to have infinitely many solutions.
Solved.
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