x 2y 3z 1 2x 5y 3z 1 x bz c Find all values of b and

x + 2y + 3z = 1

2x + 5y + 3z = -1

x + bz = c

Find all values of b and c (if they exist) such that all three planes are non-coincident and parallel

For three planes to be parallel, their normals must be parallel

n1 = normal of plane 1
n1 = <1 , 2 , 3>

And n2 = <2 , 5 , 3>

And n3 = <1 , 0 , b>

Notice n1 and n2 are not multiples of each other.

So, since n1 , n2 and n3 are not all multiples of each other,
we can safely say that there is no way these planes are parallel and non-coincident

there are no possible values of b and c