Kirchhoff's Rules

ℰ₁= V,
r₁= Ω,
R₁= Ω,
ℰ₂= V,
r₂= Ω,
R₂= Ω,
R₃= Ω,

(r₁ + R₁ + R₃)(r₂ + R₂ + R₃)-R₃²= Ω²
ℰ₁(r₂ + R₂ + R₃)-ℰ₂R₃=VΩ
(r₁ + R₁ + R₃)ℰ₂-R₃ℰ₁=VΩ
I₁= A, A
I₂= A, A
Explanation, Derivation of Formulas:
I₁r₁ + I₁R₁ + I₃R₃ = ℰ₁,
I₂r₂ + I₂R₂ + I₃R₃ = ℰ₂,
I₃ = I₁ + I₂,

I₁r₁ + I₁R₁ + (I₁ + I₂)R₃ = ℰ₁,
I₂r₂ + I₂R₂ + (I₁ + I₂)R₃ = ℰ₂,

I₁r₁ + I₁R₁ + I₁R₃ + I₂R₃ = ℰ₁,
I₂r₂ + I₂R₂ + I₁R₃ + I₂R₃ = ℰ₂,

I₁(r₁ + R₁ + R₃) + I₂R₃ = ℰ₁,
I₁R₃ + I₂(r₂ + R₂ + R₃) = ℰ₂,

ax+by=p
cx+dy=q

a b x p
c d y q

x=(pd-qb)/(ad-cb)
y=(aq-cp)/(ad-cb)

I₁ = {ℰ₁(r₂ + R₂ + R₃)-ℰ₂R₃}/{(r₁ + R₁ + R₃)(r₂ + R₂ + R₃)-R₃²}
I₂ = {(r₁ + R₁ + R₃)ℰ₂-R₃ℰ₁}/{(r₁ + R₁ + R₃)(r₂ + R₂ + R₃)-R₃²}