Kirchhoff’s Rule-Current in a Two Loop Circut

1. The problem statement, all variables and given/known data

Find the current in each branch of the circuit of the figure below . Specify the direction of each.

The top resistance is 22, middle is 56, and the bottom is 75.

I’m predicting the current is going in this direction:

2. Relevant equations
—Kirchoff’s Rules (Junction rule & Loop Rule)

3. The attempt at a solution

My starting/reference point is at the 5V battery on the positive anode side; going toward the right.

EQ1: -5v – 22i1 + 56i2 = 0

EQ2: -5v – 22i1 + 75 i3 + 1v = 0

EQ3: i2= i1 + i3

The problem I’m having is the fact I have two variables to solve for; regardless how I manipulate. the equations.

For instance:

if you solve for i1 in EQ2: i1= (-4v + 73 i3)/ 22

….. now I plugged i1 into EQ1: -5V – 22 ((-4v + 73 i3)/ 22) + 56 i2 = 0

-As you can see, I still have two unknowns: i3 and i2. The Junction rule eq (EQ3) wouldn’t help either. It would just replace i3 with i1 again. So with that said, what am I doing wrong/missing? I have all 3 loops covered in my two equations, so I’m kind of lost. Any help would be appreciated.

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