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Question 1136394: (a) Consider the following system of linear equations:
x + 5y + z = 0
x + 6y − z = 0
2x + ay + bz = c
Find values of a, b, and c such that the above system of linear equations has:
(i) exactly one solution;
(ii) an infinite number of solution;
(iii) no solution;
Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website! .
Add the first and the second equations. You will get
2x + 11y +0*z = 0. (4)
The third equations is
2x + ay + b*z = c (3)
Comparing (3) and (4), you can conclude:
(iii) if a= 11, b= 0 and c is any nonzero number, then the given system has no solution.
{ii) if a= 11, b= 0 and c= 0, then the system has infinitely many solutions.
(i) if a= 11, b=/= 0 then the system has a unique solution for any value of c.
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Notice that the problem asks to find the value "a", "b" and "c" satisfying some conditions - and I found such values "a", "b" and "c".
The problem does not require to find ALL values of "a", "b" and "c", satisfying the given requirements.
Therefore, I even didn't try answer this EXTENDED question.
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