Circuit with 3 batteries and 4 resistors

[icesalmon]

Homework Statement

The question pertains to the first problem from this link
https://courses.physics.illinois.edu/phys212/sp2014/practice/practice.pl?exam2/fa10
They ask for the value of "I" in terms of V1, V2, R1, and R2.

Homework Equations

V = IR
∑V_i = 0 [/B]

The Attempt at a Solution

Choosing Kirchoff's Voltage Law will produce three equations...[/B]

1). V₁ - i₁R₁ - V₂ - i₂R₁ - i₁R₂ = 0

2). V₂ - V₂ - i₃R₁ + i₂R₁ = 0

3). V₁ - i₁R₁ - V₂ - i₃R₁ - i₁R₂ = 0

The question makes it seem as if I could solve for that current with just one equation. I'm confused on how to label the currents given the positioning of the battery terminals in the circuit as well. They seem to be pumping current directly into each others positive terminals, why is this an efficient set up for the circuit in the first place? Should I combine the two V₂ batteries because voltages in parallel are equal?

 

[icesalmon]

okay so just messing with these equations, I₃ = I₂ and (1) and (3) become the same exact thing, working with just one equation
V₁ - i₁R₁ - V₂ - i₂R₁ - i₁R₂ = 0

 

[mfb]

Check the current sum at one of the nodes. The sum of incoming currents is equal to the sum of outgoing currents.

 

[icesalmon]

I'm sorry, I completely forgot my node equation. I found it quite easily after that third equation. Thank you

 

[CWatters]

They seem to be pumping current directly into each others positive terminals, why is this an efficient set up for the circuit in the first place?

Exam questions don't have to be representative of real world problems but you do come across circuits where one voltage source delivers current to another like this. I bet you have at least one battery charger at home. If you have one that charges several loose cells it's not to dissimilar.