Question

suppose the system

2x+4y+3z=f

x+2y-3z=g

x+2y+cz=h

Find a relation (if possible) between f,g,h,c,d such that the system is inconsistent and consistent. Can we find a relation which gives a unique solution, infinite many solution? Justify your answer.

Answer 1

First note the similarity between the second and third equations.

We can make them inconsistent by putting 

$c=-3$ and $g\neq h$

Next note that if we put $c=-3$ and $g= h$

then the second and third equations are consistent with one another and with the first equation.

Finally note that all three equations can only constrain the value of $x+2y$ , not $x$ or $y$

 individually. So when the system does have a solution, it has infinitely many solutions.