Example 1 - Solving a 3x3 system of equations using determinants

Example:
Find I₂ only, using Cramer's Rule.
I₁ + I₂ + I₃ = 0
6 I₁ = 8 + 10 I₃
6 I₁ - 2 I₂ = 5

Solution:
Start by rearranging (standardizing) the three equations:
rearrange:
1 I₁ + 1 I₂ + 1  I₃ = 0
6 I₁ + 0 I₂ - 10 I₃ = 8
6 I₁ -  2 I₂  + 0 I₃ = 5

Want to find I₂ using Cramer's Rule...
I₂ = ∆₂  / ∆

∆ =

1	1	1	1	1
6	0	-10	6	0
6	-2	0	6	-2

∆ = (1) (0) (0) + (1) (-10) (6) +(1) (6) (-3)
     - (1) (0) (6) - (1) (-10) (-2) - (1) (6) (0) 
.......
∆ = -92  <--- System determinant

∆₂ =

1	0	1	1	0
6	8	-10	6	8
6	5	0	6	5

∆₂ = (1) (8) (0) + (0) (-10) (6) +(1) (6) (5)
     - (1) (8) (6) - (1) (-10) (5) - (0) (6) (0)
∆₂ = 32

Finally: I₂ =  ∆₂ / ∆ = 32 / -92 = -8 / 23
I₂  » -0.35

Note:
If desired you could find I₁ & I₃
∆₁ =

0	1	1
8	0	-10
5	-2	0

∆₁ = ... = -66

∆₃ =

1	1	0
6	0	8
6	-2	5

∆₃ = ... = 34

\ I₁ = ∆₁ / ∆ = -66 / -92 

I₁ » 0.72

\ I₃ = ∆₃ / ∆ = 34 / -92 

I₃ » -0.37