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I am working on the following circuit:
enter image description here
I am asked to find the node voltages at N1, N2 and N3. This problem seems pretty straight forward to me but the challenge I'm facing is due to resistor R4. Resistors R4, R3 and R5 seem to form a delta. I have already attempted this problem by solving all the currents, however, my mesh calculations seem to be wrong as the currents I got don't give me the correct voltage drops. So my question here is, should I perform a Delta to Y transform or just solve the circuit as it is.

Here are the general mesh equations I came up with. I am writing them down generally as it is not the main issue I have with this problem but I am doing so to show what I have done so far. Oh and, on the original problem, it only gave the current directions i1 to i3. I added the current i4 which I think should be in between R2 and R4:
$$Mesh1$$ $$-12+i_1(R1)+(i_1-i_4)R2=0$$ $$Mesh2$$ $$R3(i_2)+R5(i_2-i_3)+(i_2-i_4)R4=0$$ $$Mesh3$$ $$R6i_3+R7i_3-2+(i_3-i_2)R5=0$$ $$Mesh4$$ $$(i_4-i_2)R4+(i_4-i_1)R2=0$$

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    \$\begingroup\$ R4 is in parallel with R2. Just replace R2 with the parallel resistance. \$\endgroup\$ – John D May 2 at 14:41
  • \$\begingroup\$ But it can be parallel to R5 right? \$\endgroup\$ – AugieJavax98 May 2 at 14:42
  • \$\begingroup\$ Notice that R4 is connected between node1 and gnd exactly the same way as R2. Hence they are connected in parallel. \$\endgroup\$ – G36 May 2 at 14:42
  • \$\begingroup\$ Is the R5 resistor is connected between node 1 and GND? \$\endgroup\$ – G36 May 2 at 14:44
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    \$\begingroup\$ R5 is not in parallel with R2||R4 because R5 is connected to a different node then R2||R4. \$\endgroup\$ – G36 May 2 at 14:59
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I've re-drawn your schematic, as follows, and I've kept what I believe to be your current assignments:

schematic

simulate this circuit – Schematic created using CircuitLab

I didn't bother to convert \$R_2\$ and \$R_4\$ into a parallel equivalent, since I think you may have been required not to do that and to set up a sufficient number of loop currents to solve it. \$I_1\$ could also have been re-directed along a different path, if you'd have wanted. But your choice is fine, since all the loops you've chosen do interact with each other in some fashion. So the above loop currents should be sufficient for a solution.

I'm a little bit bothered by the values for \$R_6\$ and \$R_7\$ as I've written them in the schematic. Your writing seems to suggest those values, but I'm uncertain. But the equations won't matter. So right or wrong, it's fine. You can adjust as needed.

(And you should learn to use the built-in schematic editor that this site provides you. It will help you provide numbered parts and greatly improve the readability of your questions.)

The resulting mesh equations are:

$$\begin{align*} V_\text{A} - I_1\cdot R_1 - \left(I_1-I_4\right)\cdot R_2&=0\:\text{V}\\\\ 0\:\text{V}-\left(I_4-I_1\right)\cdot R_2-\left(I_4-I_2\right)\cdot R_4&=0\:\text{V}\\\\ 0\:\text{V}-\left(I_2-I_4\right)\cdot R_4-I_2\cdot R_3-\left(I_2-I_3\right)\cdot R_5&=0\:\text{V}\\\\ 0\:\text{V}-\left(I_3-I_2\right)\cdot R_5-I_3\cdot R_6-I_3\cdot R_7&=V_\text{B} \end{align*}$$

With \$V_\text{A}=+12\:\text{V}\$ and \$V_\text{B}=+2\:\text{V}\$, these should solve out for the four currents and from there you can apply them to find the node voltages, as well.

Check your equations against mine.

Note that there is no need to perform \$Y-\Delta\$ or \$\Delta-Y\$ conversions or perform a parallel equivalent for \$R_2\$ and \$R_4\$.

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