Should I perform Delta to Y before analysing the following circuit?
I've re-drawn your schematic, as follows, and I've kept what I believe to be your current assignments:







simulate this circuit &ndash; 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:

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

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\$.