To solve for $x$, $y$, and $z$ in this non-right triangle, let's analyze the triangle’s angles and sides using trigonometric laws.

Given:

• Side opposite angle $y^\circ$ = 4
• Side opposite angle $x^\circ$ = 9
• The known angle inside the triangle = $54^\circ$

Steps to Solve:

1. Find Angle $y$ Using the Sum of Angles in a Triangle: Since the sum of angles in any triangle is $180^\circ$, $x + y + 54^\circ = 180^\circ$ Rearranging, we get: $x + y = 126^\circ$

2. Apply the Law of Sines: Using the Law of Sines, which states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ we can relate the sides and angles of the triangle.

3. Set Up Equations and Solve: Since we have two unknown angles, $x$ and $y$, and the side lengths are given, let's solve these equations by using the Law of Sines and substitute values to get approximate answers.

I’ll go through the calculations and get back with the results.It appears there was an issue with solving for the angles directly. I’ll refine the approach and re-calculate by explicitly isolating possible values for $x$, $y$, and side $z$ using numerical techniques.

Let me resolve this calculation and get back with the correct values.It seems the calculations didn’t yield a valid solution for $x$, $y$, and $z$ under the given setup. This might indicate an inconsistency in the triangle's side lengths and angles that makes solving for these values challenging with typical trigonometric methods.

Could you please confirm the side lengths and angles, or let me know if there’s any other detail I should consider?