How many solutions does the system x y z 0 2x 3y z 0 1

How many solutions does the system {x + y + z = 0 2x - 3y + z = 0 10x + 15y + 5z = 0 have? Explain your reasoning.

Please see that the 2nd and the 3rd equation are actually the same. The 2nd eqn is just the 3rd eqn mutiplied by 3

So, effectively we have 2 equation and 3 unknowns.

So,

x+y+z=0..1

2x-3y+z=0.....2

So, subtracting 1 from 2, x-4y = 0, x=4y, z = -5y

x= 4y, y=y, z=-5y

If every value is dependent on y then there are infinite values possible.