If you split it into sections it should be easier than combining it into one.

If you have $F_1$ , $F_2$, $F_3$ as the tensions of the strings

$$
F_{1x} = F_{2x}
$$
$$
F_{2x} = F_{3x}
$$
$$
F_{1y} + F_{2y} = 12g
$$
$$
F_{2y} + 7g = F_{3y}
$$

Then you can use trigonometry and then eliminate.

Edit:
Eliminate $F_{2y}$

$$
F_{1y} + F_{3y} = 19g
$$

Then Trig
$$
F_1sin(45) + F_3sin(30) = 19g
$$
Along with
$$
F_1cos(45) = F_3cos(30) 
$$