Prove there are no non-trivial solution to $3x^2 - 5y^2 + 7z^2 = 0$

Question

I've tried using modulo $3$, and I get it down to

$y^2 + z^2 = 0 \pmod 3$ ; I don't know where to go from here though.

I justified my answer by stating that, because we're in $\pmod 3$ and we need non-trivial solutions, the only solutions possible are $y = 1,2$ and $z = 1,2$; all of which will not give us $0$.

Thanks!

Answer 1

Modulo $4$, this equation is $3x^2+3y^2+3z^2\equiv 0\pmod 4$, or $x^2+y^2+z^2\equiv 0\pmod 4$. The only squares modulo $4$ are $0$ and $1$, so conclude that $x,y,z$ must all be even.

Answer 2

Hint: $y^2+z^2 = 0\pmod 3 \to y=z=0\pmod 3$