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 $$ F_1cos(45) = F_3cos(30) $$

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 $$ F_1cos(45) = F_3cos(30) $$

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.

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 $$ F_1cos(45) = F_3cos(30) $$