Let's say I have an easy functions such as:
$$x_1 x_2 x_3 = 5$$
Then, I can express each variable as a function of others:
$$x_1 = \frac{5}{x_2 x_3}, \qquad x_2 = \frac{5}{x_1 x_3}, \qquad x_3 = \frac{5}{x_2 x_1}$$
Which can be plotted with ease. BUT if I have more sophisticated function such as:
$$\left( \left( x_1^{-3} + x_2^{-3} \right)^{\frac{-4}{-3}} + \left( x_2^{0.5} + x_3^{0.5} \right)^{\frac{-4}{0.5}} \right)^{\frac{1}{-4}} = 10$$
I am not able to express one variable as a function of others.
Therefore, how, in general, are functions like these even plotted? What mechanisms are being used? What is the general intuition if one wants to plot this?