Let $f_1,f_2,f_3$ be continuously differentiable functions from $\mathbb R^4$ to $\mathbb R$. Give sufficient conditions so that the equations $f_1(x,y,z, t)=0$, $f_2(x,y,z, t)=0$, $f_3(x,y,z, t)=0$ can be solved for x, y, z in terms of t.
My attempt: Let $f_1,f_2,f_3:\mathbb R^4\rightarrow \mathbb R$
Let $f:\mathbb R^4\rightarrow \mathbb R^3$ defined by $f=(f_1,f_2,f_3)$, then f is a smooth function. The differential matrix
$Df=\left[\begin{matrix}\dfrac{\partial f_1}{\partial x}&\dfrac{\partial f_1}{\partial y}&\dfrac{\partial f_1}{\partial z}\\\dfrac{\partial f_2}{\partial x}&\dfrac{\partial f_2}{\partial y}&\dfrac{\partial f_2}{\partial z}\\\dfrac{\partial f_3}{\partial x}&\dfrac{\partial f_3}{\partial y}&\dfrac{\partial f_3}{\partial z}\end{matrix}\right]$
I do not know how to step up from this. Please, help.