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I need to solve this problem as part of my review in college physics. The task is to find the value of 5 branch currents using Kirchhoff's laws and elimination method (or maybe called elimination by substitution BUT not using any matrix method).

The circuit is: enter image description here

I got the KCL equations:

a: $I_1 - I_2 - I_3 = 0$

b: $I_3- I_4- I_5 = 0$

and the KVL equations:

$\text{Loop 1}: 15 - 4I_1 - 6I_2 = 0$

$\text{Loop 2}: 6I_2 - I_3- 3I_4 = 0$

$\text{Loop 3}: 3I_4 - 2I_5 - 10 =0$

My problem is I can't get the correct answer by elimination method. Please help me to provide the solution if the given answers are:

$I_1 = 1.89$
$I_2 = 1.24$
$I_3 = 0.65$
$I_4 = 2.26$
$I_5 = -1.61$

Please show the solution and give tips on how to solve this type of problems using elimination method so other students can benefit and learn here.

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closed as off-topic by jinawee, centralcharge, Brandon Enright, Waffle's Crazy Peanut, Chris White Feb 2 '14 at 8:19

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It looks to me that you setup the loop equations correctly. I substituted your KCL equations for $i_2$ and $i_4$ into the loop equations so that there was only $i_1$, $i_3$ and $i_5$ to solve for, and I was able to get the correct answers that you gave.

enter image description here

I also checked these answers with a PSpice simulation, and they agree with your answers as well as the matrix solutions above.

enter image description here

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We got $5$ linear equations and $5$ variables. There are many ways to solve it but i'll use Gaussian elimination as you asked. It's all about mathematics.

Tip: Use Cramer's method, it is the easiest.

$1i_1+(−1i_2)+(−1i_3)+(0i_4)+(0i_5)=0 \tag{1}$ $0i_1+(0i_2)+(1i_3)+(-1i_4)+(-1i_5)=0 \tag{2}$ $-4i_1+(−6i_2)+(0i_3)+(0i_4)+(0i_5)=-15 \tag{3}$
$0i_1+(−6i_2)+(−1i_3)+(-3i_4)+(0i_5)=0 \tag{4}$ $0i_1+(01i_2)+(0i_3)+(3i_4)+(-2i_5)=0 \tag{5}$

In matrix form taking the respective coofficients
image 0

Tip:The best way to solve by elimination is convert the matrix into an upper triangular matrix or in echolon form.

image 1
image 2
image 3
Which is the correct solution.

Last tip: Do the rough work at some other page and write each reduced matrix on another plain page.

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