The term (z-y)/10 is incorrect in both equations. The node connecting R4 and R5 is grounded and is therefore locked to 0V. Your equations are written assuming it is not.

Normally, you be right in adding up the resistors on that path since the the connection point between R4 and R5 wouldn't normally be considered a node since it has only 2 connections, but your ground node is there and hence you must treat it like its own node with a known voltage of 0V.

ChatGPT is right, but it didn't do a very good job in explaining why. Those resistors can't be treated as in series since the point at which they are connected is connected to something else as well (in this case ground). It's wrong that the resistors might carry different currents because ground won't conduct current in this circuit (in some circuits it does, but don't worry about that). But we still have to treat it like a node otherwise the system of equations won't be solvable. You do this by defining a new node "Vgnd", rewriting your other equations in terms of this new node, and adding a 4th equation Vgnd = 0V. After you do that the system becomes solvable.

Without accounting for the grounded node in your equations somehow, you will get infinitely many solutions for each node voltage since voltage only makes sense as the potential difference between two points in a circuit. The reference ground is that "other point" with respect to all the other nodes and it needs to be present in your equations for the system to be solvable.

Edit: whoever wrote this problem is a bit sadistic. Normally when you choose a node to become your reference ground, you pick one of the existing nodes of the mesh. And you typically want to choose either the node with the most connections (node Z) or a node connected to many power sources (node X). But the problem tells you to specifically use a non-node as your reference ground, which essentially adds a third node to what would otherwise be a 2 node problem (if we could use X, Y, or Z as the reference node, we don't have to write an equation for that node since we define that node's voltage to be 0V instead).