System of equations problem

Hi MHF,

Here is an exercise and I've done part a) and b) but not part c). So I'd like a check up for the 2 first parts and a tip for the last part. Thanks in advance.

Consider the system of equations given by , , .

a)Show that the system has at least 2 solutions that are linear independent if .

My attempt : I wrote the system of equations as a matrix and reduced it in order to get . From which we get so any vector in the plane can be written as . Let and , we get that a solution is , let and and we get which are clearly 2 linear independent solutions.

b)Determine the values of a, b and c such that the system has solution.

My attempt : I wrote the system as an augmented matrix (augmented with the column a, b, c) and finally reaches the same matrix as before of course, but with the column vector from which we have so , is any value and .

c)Show all the solutions for , and , if there are/is.

My attempt : so the solutions are infinitely many? All numbers , , and such that are solutions?