Kirchoff's Voltage Law transposing problem

[JKhim]

Homework Statement

So guys, its good to be back here and ask some of your opinions. Im new in this community, so i hope you'll guide me for a good start. Thank you

So here is my problem about Kirchoff's Voltage Law
When I substitute all the givens in the equation, i got confused because the notes i got from my classmate and my source in the internet is different on how it tranposed.

So guys, I will send a picture of the circuit given problem, and the solution i got using the notes of my classmate that he got in our physics teacher because that time, i was absent for certain reasons, and also the solution i got using the information in internet, it has many comments and reactions so i looked up to it and use it as a guide or basis too.

Feel free to ask about my work guys

Relevant Equations

V1-I1R1-I2R2-V2=0

 

[JKhim]

To be clarify:
The calculation i got first is this:
V1-I1R1-I2R2-V2=0
5I1+10I2=6V+6V
5I1+10I2=12V
0.42I1+0.83I2=1
Second:
V1-I1R1-I2R2-V2=0
6V-5I1-10I2-6V=0
5I1+10I2=0
Which is correct guys? i feel sorry to myself because i got this trust issue in my doings, and I am just asking about your opinions because i know you can help me to clarify things, because this little confusion can affect all my other answers, and it might lose points in my assignment. I hope you will help me

 

[cnh1995]

Your I1, I2 and I3 are all flowing away from the middle junction. As per KCL, their sum should be zero.

 

[JKhim]

Your I1, I2 and I3 are all flowing away from the middle junction. As per KCL, their sum should be zero.

So you're saying that my kcl is wrong, instead of I2=I1+I3, it should I1+I2+I3=0 and by transposing it we get -(I1+I3) = I2, is that correct?

 

[cnh1995]

You need 3 independent equations here as you have three variables I1, I2 and I3.

So you're saying that my kcl is wrong, instead of I2=I1+I3, it should I1+I2+I3=0 and by transposing it we get -(I1+I3) = I2, is that correct?

Yes.
That is one of the 3 equations.

Can you find the other two?
Use KVL in the two loops.

 

[JKhim]

I think i got the right kvl when you say that all the currents in the problem are moving away, because i always think that these directions are just a design and not to include in a certain circuit problem, i used to follow the notes of mine even if it has a different directions, so i think this is the right equation to be used since it has the opposite direction i have in my notes.
Loop 1
V1+I1R1-I2R2-V2 = 0
Loop 2
V3+I3R3-I2R2-V2 = 0

 

[JKhim]

Is my doings correct sir?

 

[cnh1995]

V1+I1R1-I2R2-V2 = 0
Loop 2
V3+I3R3-I2R2-V2 = 0

Yes, now you can solve the three equations to get the three currents.