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