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I need to solve the circuit shown below:

schematic

simulate this circuit – Schematic created using CircuitLab And here is my attempt. My attempt

I'm sure my attempt is wrong here but I'm not really sure why.

I've tried to use the mesh method to determine the currents in the cicuit. I've ended up with these equations for the currents:

$$ i_A = i_1\\ i_B = i_2 - i_1\\ i_C = i_2 - i_3\\ $$ And KVL: $$ 1:\; 9V - 10k * i_A + 20k*i_B - 10V = 0\\ 2:\;10V - 20k * i_B - 30k * i_C + 12V = 0\\ 3:\;12V + 40k * i_C - 30k * i_C = 0 $$

Based on these equations: $$ i_A = -766\mu A\\ i_B = -330\mu A\\ i_C = -370\mu A $$ I've tested this cicuit in a simulator (falstad) and the current on R1 should be 420μA, which points to a really big mistake on my account. Any help or tip how to fix these equations would be welcomed.

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You are doing the mesh method wrong, specifically the currents on it. To do it correctly:

  1. Define the current meshes. You can select on each one does it flow clockwise or counter clockwise but as a beginner it's probably the easiest to use same rotation for all of them. enter image description here
  1. Now do the equations, correct ones are below. If there is another current flowing on same wire, you need to take that to account!

$$ \begin{cases} 10k \cdot Ia + 20k \cdot (Ia-Ib) + 10V - 9V = 0 \\ 20k \cdot (Ib-Ia) + 30k\cdot (Ib-Ic) - 12V - 10V = 0 \\ 30k \cdot (Ic-Ib) + 40k\cdot Ic + 12V = 0 \end{cases} $$

  1. Solve the equations with calculator (or by hand...) and you'll get the following results: $$ \begin{cases} Ia = 420μA \\ Ib = 680μA \\ Ic = 120μA \end{cases} $$

  2. Now you can get the real currents from pic: $$ \begin{cases} IR1 = Ia = 420μA \\ IR2 = Ib - Ia = 260μA \\ IR3 = Ib - Ic = 560μA \\ IR4 = Ic = 120μA \end{cases} $$

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