I don't understand where you got your equations from TBH.... Maybe if I pulled out a pencil and paper

But I'd do this:

Your first 3 eq are correct.

So loop 1 is:

V(source1)= V1+V2

10=2i1+ 8i2

Note the DIRECTION of the voltages in each resistor. They both oppose the source so they're both positive

Loop 2 is

V(source)=-V2+V3

The "-" before the V2 is critical. This is written because that voltage (as defined) ADDS to the voltage source

6=-8i2+4i3

There's a 3rd loop that runs around the entire perimeter. You can write an eqation for that too...

V(source1)=V2+V3-V(source2).... again the (-) is key here, because of the direction of the voltages defined

Does this help?

If not, post more of your work. Just posting the final equations doesn't help find the problem....

I don't understand where you got your equations from TBH.... Maybe if I pulled out a pencil and paper

But I'd do this:

Your first 3 eq are correct.

So loop 1 is:

V(source1)= V1+V2

10=2i1+ 8i2

Note the DIRECTION of the voltages in each resistor. They both oppose the source so they're both positive

Loop 2 is

V(source)=-V2+V3

The "-" before the V2 is critical. This is written because that voltage (as defined) ADDS to the voltage source

6=-8i2+4i3

If it's clearer, look at it as Kirchoffs "The voltages must equal zero"

V(Source2)+V2-V3=0.

V3 in that form opposes the loop, so it's negative.

There's a 3rd loop that runs around the entire perimeter. You can write an eqation for that too...

V(source1)=V2+V3-V(source2).... again the (-) is key here, because of the direction of the voltages defined

So you now have 3 equations with 3 unknowns.

Does this help?

If not, post more of your work. Just posting the final equations doesn't help find the problem....