Edit I was wrong. Look at Floris's answer.
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 $$ F_1cos(45) = F_3cos(30) $$