Real Triads of Solutions

Shew that if $a$, $b$, $c$ are positive and unequal, the equations $ax+y+z=0$, $x+by+z=0$, $x+y+cz=0$ give three distinct triads of real values for $x$, $y$, $z$; and the ratio of the products of the three values of $x$ and $y$ is $b(b-c):a(c-a)$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Miscellaneous Examples, No. 298