Determine I_a, I_b, and I_c

KVL around mesh 1.

3 I₁ + 45 I₁ - 45 I₂ + 2 I₁ - 40 = 0

50 I₁ - 45 I₂ = 40 (Eq. 1)

KVL around mesh 2.

4 I₂ - 64 + 1.5 I₂ + 45 I₂ - 45 I₁ = 0

-45 I₁ + 50.5 I₂ = 64 (Eq. 2)

Solving the two equations yields:

I₁ = 9.8 A

I₂ = 10 A

Thus

I_a = I₁ = 9.8 A

I_b = I₁ - I₂= - 0.2 A

I_c = - I₂ = - 10 A