Static equilibrium/tension force problem!

you have 4 unknowns and 3 equations which means you should be able to solve for each variable fairly easily. ( I say 4 unknowns because T3x cand T3y can both be expressed in terms of T3, with some triangular thinking.. Not sure if your torque equation is right or wrong(haven't done that stuff in a while)

 

Yes, you can and you should. [I assume you mean T_1y = (3.0 kg) x g, and T_3y = (2.0 kg) x g]

And it's important to understand why you can. So I encourage you to think about it for a couple of minutes.

Consider a tiny point on the string, right at the intersection of the T₁ string, the horizontal T₂ string, and the vertical string holding up the 3.0 kg mass. According to this small point of string, there are only 3 forces acting on it. So you can isolate this particular point to be a function of only these 3 forces. How many forces are acting on that particular point in the y-direction? Since nothing is accelerating, what does it tell you about those forces that exist in the y-direction (as seen from that particular point)?

[Edit: Btw, The above applies because we are dealing with light strings. If we were dealing with heavy rods, or rigid rods that clamped onto each other and held firm, things might be a little different.]