I am trying to find all the intersecting integral points $(x,y,z)$ of the plane $$3x+5y+4z=45$$ and the one-sheeted paraboloid $$z^2+xz=15$$.
I noticed that $$x=4t-(9-y)$$ $$z=-3t+2(9-y)$$ So I replaced $x,z$. I ended up with the hyperbola $$9 t^2 + 16 t y - 108 t + 5 y^2 - 81 y + 309 = 0$$ which I solved to find infinitely many integral solutions including $$(2,7,1)$$. My question is: Is there any easier method to find all those points knowing a common intersecting integral point?