In the given network, find the values of currents I1, I2 . . .

Question : In the given network, find the values of currents I₁, I₂ and I₃.

Doubt by Anushka

Solution : 

Applying Kirchhoff's Junction Rule at D
I₁=I₂+I₃ — (1)

Applying Kirchhoff's Loop Rule in closed loop 1
1I₁+2I₂+2I₁=2-1
3I₁+2I₂=1
3(I₂+I₃)+2I₂=1 [Using eq (1)]
3I₂+3I₃+2I₂=1
5I₂+3I₃=1 — (2)

Again, applying Kirchhoff's Loop Rule in closed loop 2

3I₃+10I₃-2I₂=3-1
13I₃-2I₂=2
-2I₂+13I₃=2 — (2)

Solving equations (1) and (2) by elimination method

[ 5I₂+3I₃=1   ]×2

[-2I₂+13I₃=2 ]×5

10I₂+6I₃=2 — (2)
-10I₂+65I₃=10 — (3)
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0 + 71I₃ = 12
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I₃=12/71 A

Putting this value of I3 in equation (2)
10I₂+6(12/71)=2
10I₂+(72/71)=2
10I₂=2-(72/71)
10I₂=(142-72)/71
10I₂=70/71
I₂=7/71 A

Putting these values of I2 and I3 in equation (1)

I₁= (7/71)+(12/71)
I₁=19/71 A

Hence, I₁ = 19/71, I₂ = 7/71, I₃=12/71