First notice that the triangle with sides \(x,y,a\) and the triangle with sides \(z,a+b,x\) are similar. This is true because the angle between sides \(x,y\) in the smaller triangle is clearly \(30^\circ\), while the angle between sides \(y,z\) in the larger triangle is clearly \(60^\circ\). So the triangles are similar with sides \(x,y,a\) corresponding to \(a+b,z,x\), respectively.
Now both triangles are \(30^\circ-60^\circ-90^\circ\), which means there's a convenient ratio between its sides. If the length of the shortest leg is \(\ell_1\), then the length of the longer leg is \(\ell_2=\sqrt3\ell_1\) and the hypotenuse has length \(\ell_3=\sqrt{{\ell_1}^2+{\ell_2}^2}=2\ell_1\).
Since \(x\) is the shortest leg in the larger triangle, it follows that \(a+b=2x\), so \(a+b=15=2x\implies x=\dfrac{15}2\)
