Three purchases of pens are recorded as follows:
• 2 ball-point pens, 1 gel pen and 3 marker pens cost Rs. 50.
• 4 ball-point pens, 2 gel pens and 6 marker pens cost Rs. 100.
• 6 ball-point pens, 4 gel pens and 9 marker pens cost Rs. 145.
Let Rs.\,x, Rs.\,y, and Rs.\,z be the respective prices of one ball-point pen, one gel pen and one marker pen.
a. (i) Examine whether a set of prices (x,y,z) exists that satisfies all three transactions.
b. (ii) State the nature of the system of equations (no solution, unique solution or infinitely many solutions) and justify your answer briefly.