please help me solve this problem below;

Question

Consider the following system of linear equations: Find values of a, b, c such that the above system of linear equations has:

(i) exactly solution;

(ii) an infinite number of solution;

(iii) no solution;

x + 5y + z = 0

x + 6y − z = 0

2x + ay + bz = c

Answer 1

a=10, b=2, c=0 for an infinite number of solutions

as that makes the 3rd equation just twice the 1st and linearly dependent

make one equation a contradiction of another, to get zero solutions.

a=10, b=2 c= 1

for a unique solution, just have all 3 equations be linearly independent

and no contradiction, no linear dependence. which is just about any

other choice of values for a, b and c.