So if we go to loop 4, our mesh 4 and

do the same thing, we can sum the voltages up just like we did before.

So, the first voltage drop is across 1 kilo ohm resistor,

and it's 1k I4 minus I3 plus voltage

drop across 1 kilo ohm resistor in the center

of the circuit, 1k, I4 minus I sub 2.

And then we have the voltage drop from the voltage source, which is 12.

And that is equal to zero.

So they're for loop or mesh equations.

We know what I1 and I2 are, I2 in terms of another variable V sub x.

So we have one unknown or

one variable which I1, two fro I2,

three for V sub x, four for I sub 3,

five for I sub 4 and six for I sub x.

So we need six equations.

So we need to have find two more equations that relate our variables to one another.

That's means the independent of the prior four.

So we can investigate those first of all by looking first at V sub x.

What is V sub x?

It's the voltage drop across the 2 kilo ohm resistor.

So we can write an equation V sub x is

equals to 2K I sub 3 minus I sub 1).

Because I sub 3 flows into the positive polarity of the voltage drop