SOLUTION: (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 equation

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.

------------

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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