Can we solve for x and y? All I see is algebra here, is my intuition wrong?

More directly than the other comments: No you can’t solve for x and y here and yes your intuition is wrong.

These are functions. I don’t know your level of knowledge in math or programming and what that would mean to you. Here’s an example.

double(x) -> x*2

So, double(3) = 6. You can’t solve for x because x doesn't have a value. It’s a placeholder for whatever you put in.

These combinators are functions that take other functions and return them unmodified. “Unmodified” is a little misleading because it can do things like drop inputs.

The intuition here is that combinators are higher order functions which take functions and combine them together in various ways. So for a simple example "fix" is a combinator in regular maths where

Fix f = {f(x): f(x) = x for all x in the domain of f}

So if f is a function or a group action or whatever, the fixed-point set of f is all points x in the domain of f such that f(x)=x. ie the points which are unchanged by x. So if f is a reflection, the points which sit on the axis of reflection.

The fixed-point combinator is of particular relevance to this site because it's often called the y combinator.

It's more like a recipe (for functions).

The first example, I, is an identity function. It takes y and returns y.

The second, K, is a constant which takes X and y and returns x.

This gets more complicated as you go along. The idea is that you get rid of a lot of the syntax for composition and have it all be implicit by what you put next to each other (given APL programs are usually one long line of a bunch of different symbols all representing functions).

Combinators can be a bit sill for values. The usefulness come when you use them as a meta language for functions.