Solving Currents in Electrical Networks for Graph Theory

Question

I need help trying to solve this question, been cracking my head for the whole week and my professor said he used an online solver but in exams we have to solve by hand!

Given these 8 equations, we are supposed to solve for $i_0, i_1, \dots, i_7$: $$\begin{array}{rl} i_1+i_2 &= 12 \\ i_2+i_5+i_6 &= 0 \\ i_3+i_5+i_7 &= 0 \\ i_2-i_4+i_5+i_7 &= 0 \\ i_0-i_1 &= 0 \\ i_1-i_2-i_3+i_5 &= 0 \\ i_3-i_4-i_7 &= 0 \\ i_5-i_6-i_7 &= 0 \end{array}$$

I know the answers are: $ i_0=8, i_1=8, i_2=4, i_3=2, i_4=2, i_5=2, i_6=2, i_7=0 $, but I don’t know how to solve by hand!

Answer 1

The above equations can be represented by the following augmented matrix: $$\left(\begin{array}{rrrrrrrr|r} 0&1&1&0&0&0&0&0 & 12 \\ 0&0&1&0&0&1&1&0 & 0 \\ 0&0&0&1&0&1&0&1 & 0 \\ 0&0&1&0&-1&1&0&1 & 0 \\ 1&-1&0&0&0&0&0&0 & 0 \\ 0&1&-1&-1&0&1&0&0 & 0 \\ 0&0&0&1&-1&0&0&-1 & 0 \\ 0&0&0&0&0&1&-1&-1 & 0 \end{array}\right).$$ It’s pretty straightforward, although a bit tedious, to perform row-reduction on it to solve the system.

Answer 2

yeah you are right, it should be -2, I guess the book have some misprints. But it is too tedious for a 5 marks question in exam, don't know if there is a faster way to do it. I did an online solver and it takes 28 steps! Aint no time for that!