The system that I need to solve is $$ \begin{align} &i_1 + i_2+ i_3 &=0\\ &i_1+i_4+i_6 &=0\\ &i_5+i_6&=i_2\\ &-v_{s1} + i_1r_1 + i_3r_3 - i_4r_4 + v_{s4} &= 0\\ &-i_2r_2 + v_{s2} - i_5r_5 - i_3r_3 &= 0\\ &-v_{s4} + i_4r_4 + i_5r_5 - i_6r_6 - v_{s6} &= 0\ \end{align} $$ for the variables $i_k$.
For the asker: the question as it should appear
You can see the original question at: http://fourier.eng.hmc.edu/e84/lectures/ch2/node2.html :example 2. Please provide the steps. Thank U in advance.
For convenience, let's rewrite the system as $$ \begin{align} &i_1 &&+ i_2 &&+ \,\,\,\,i_3 &&&&&&&&=0\\ &i_1 &&&&&&+i_4&&&&+i_6 &&=0\\ &&&-i_2&&&&&&+i_5&&+i_6&&=0\\ &r_1i_1 &&&&+r_3i_3 &&-r_4i_4 &&&&&&= v_{s1}-v_{s4}\\ &&&-r_2i_2 &&- r_3i_3 &&&&-r_5i_5 &&&&= -v_{s2}\\ &&&&&&& +r_4i_4 &&+r_5i_5 &&-r_6i_6 &&= v_{s4}+v_{s6}\ \end{align} $$ (the above should look like this):
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From there, you should be able to use Gaussian elimination