Find the length of a triangle's side

As you noticed $2\tan(y)=\tan(19+y)$. Hence $$2\tan(y)=\frac{2x}{27}=\tan(19+y)=\frac{\tan(19)+\tan(y)}{1-\tan(y)\tan(19)}=\frac{\tan(19)+\frac{x}{27}}{1-\tan(19)·\frac{x}{27}}=\frac{27\tan(19)+x}{27-\tan(19)x}$$ $$\iff \frac{2x}{27}=\frac{27\tan(19)+x}{27-\tan(19)x}\iff 54x-2\tan(19)·x^2=729\tan(19)+27x$$ $$\iff 2\tan(19)·x^2-27x+729\tan(19)=0$$ This gives $$x=15.15...\lor x=24.05...$$

Now, as you can see in the following image (made with geogebra), both values are correct:

enter image description here

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Geometry

Triangles

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