Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$

Question

Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$

I thought that I can apply the implicit function theorem, but I found that $f_1(x,y,z,t)=x^2+y^2+z^2-3$, $f_1(-1,-2,1,0)\ne 0$. What can I do? Appreciate any suggestion.

Answer 1

HINT:

To apply the implicit function theorem you need to check that the jacobian $$\frac{\partial (f,g,h)}{\partial(x,y,z)}_{|(−1,−2,1,0)} \ne 0$$ where $f(x,y,z,t) = x^2 + y^2 + z^2 -3$, $\ \ g(x,y,z,t)= x y + z t - 2$, $\ \ h(x,y,z,t) = x z + y t + e^t$.