Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y,z) = (1,2, 4)$

Question

Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y, z) = (1,2, 4)$.

I am confused on how to approach this problem

Answer 1

$$y=2x;\quad z=2y;\quad z=4x.$$

Answer 2

The meaning of the question is a little unclear. If we are allowed to use constants in the equations then we trivially can write the equations $$x-1=0,\hspace{10mm}y-2=0,\hspace{10mm}z-4=0$$ to which $(x,y,z)=(1,2,4)$ is clearly the unique solution.

If we are not allowed to use constants, then there exists no homogeneous system of $3$ variables which has $(x,y,z)=(1,2,3)$ as the unique solution. This is because the form of the equations would be $$ax+by+cz=0$$ $$dx+ey+fz=0$$ $$gx+hy+iz=0$$ which already has the trivial solution $(x,y,z)=(0,0,0)$ so the solution $(x,y,z)=(1,2,4)$ could not be unique. If this is what the question intends, then it doesn't have a solution.