Circuit analysis with Z-Diode
You can create a single analytical solution but you would need a fairly sophisticated model for the zener diodes.

The usual approach is to recognize that each diode can be in reverse breakdown or not in reverse breakdown. If the diode is in reverse breakdown then it can be modeled (crudely) as an ideal voltage source at the zener voltage. If it is not in breakdown then it can be modeled as an open circuit, or as an ideal current source of 0A.

So there are four possible combinations of in breakdown or not in breakdown for these two diodes. You need to analyze the circuit for these four possible scenarios. If, in a given scenario, you assume that a diode is in breakdown but find that the reverse current is negative then you know that assumption (and that scenario) is not valid. If, in a given scenario, you assume that a diode is not in reverse breakdown but find that the reverse voltage across it is greater than the zener voltage then you know that assumption is not valid.

You should find that three of the four possible scenarios are not valid, and just one of them is valid.