Since you have the 42-deg angle and the length of the opposite side, 31, you can use the Law of Sines.
[tex] \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} [/tex]
We will use the law of sines first to find x.
Once we know x, we can find the angle opposite y.
Then we can find y.
[tex] \dfrac{31}{\sin 42^\circ} = \dfrac{37}{\sin x} [/tex]
[tex] \sin x = \dfrac{37\sin 42^\circ}{31} [/tex]
[tex] \sin x = 0.798639 [/tex]
[tex] x = \sin^{-1} 0.798639 = 53^\circ [/tex]
Let z = the angle opposite y.
z + 42 + 53 = 180
z = 85
[tex] \dfrac{31}{\sin 42^\circ} = \dfrac{y}{\sin 85^\circ} [/tex]
[tex] y = \dfrac{31 \sin 85^\circ}{\sin 42^\circ} [/tex]
[tex] y = 46.2 [/tex]
