Extra explanation: proof

Property slope tangent line to level curve: For a change of $x$ by $\Delta x$ and a change of $y$ by $\Delta y =  -\frac{z_x'(x,y)}{z_y'(x,y)}  \cdot \Delta x$, the change of $x$ multiplied by the slope of the tangent line to the level curve at $(x,y)$, the function value stays approximately the same. Or,
$$z(x+\Delta x, y  -\dfrac{z_x'(x,y)}{z_y'(x,y)} \cdot \Delta x) \approx z(x,y).$$

Proof: Changes $\Delta x$ and $\Delta y$ of the variables $x$ and $y$ such that the function value $z(x,y)$ does not change satisfy the equation
$$z(x+\Delta x,y+\Delta y)-z(x,y)=0.$$
In general it will not be easy to solve this equation for $\Delta x$ and $\Delta y$. According to the property of the partial derivatives we have
$$z(x+\Delta x,y+\Delta y)-z(x,y) \approx z_x'(x,y)\Delta x+z_y'(x,y)\Delta y$$
and therefore we approximate the exact changes. For this we take $\Delta x$ and $\Delta y$ such that
$$z_x'(x,y)\Delta x+z_y'(x,y)\Delta y=0,$$
and, hence
$$\frac{\Delta y}{\Delta x}=-\frac{z_x'(x,y)}{z_y'(x,y)}.$$
For changes $\Delta x$ and $\Delta y$ in this proportion, the function values will approximately not change.