Why Aren't My Calculated Currents Correct in This Circuit Analysis?

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Homework Statement

In the given circuit find the current i₁,i₂,i₃.
(Image of the circuit is attached)

Homework Equations

current=(emf)/total resistance

The Attempt at a Solution

The original question actually asks me to find the currents using Kirchoff's laws .Since I feel this method to be a little tedious I tried to do it another way.
But the problem is the answers are not coming correct this way ,(which is as follows )

First since the two cells are connected opposite to each other i have assumed the resultant emf of the 21V to be 4V and I have neglected the other 17V cell.
then considering 2Ω and 5Ω resistances to be parallel to each other their resultant comes to be
(10/7)Ω .And this one being in series with 3Ω, The final resistance of the circuit comes out to be
(31/7)Ω.
Accordingly current should be i₁=28/31 A
I am going seriously wrong somewhere , but can't figure out where. .
Can you please help?
Thanks in advance!

 

[Tanya Sharma]

The Attempt at a Solution

The original question actually asks me to find the currents using Kirchoff's laws .Since I feel this method to be a little tedious I tried to do it another way.
But the problem is the answers are not coming correct this way ,(which is as follows )

First since the two cells are connected opposite to each other i have assumed the resultant emf of the 21V to be 4V and I have neglected the other 17V cell.

What you have presumed would have been correct if the two batteries would have been in series .Do you think the batteries are in series ?

Please be careful ,there is a middle branch between the two batteries.The current flowing through them will not be the same .

then considering 2Ω and 5Ω resistances to be parallel to each other their resultant comes to be
(10/7)Ω .

Wrong .

And this one being in series with 3Ω

Wrong .

 

[Redbelly98]

The original question actually asks me to find the currents using Kirchoff's laws .Since I feel this method to be a little tedious I tried to do it another way.
But the problem is the answers are not coming correct this way ,(which is as follows )
.
.
.
I am going seriously wrong somewhere , but can't figure out where. .
Can you please help?
Thanks in advance!

You'll need to use Kirchhoff's rules here, tedious as they are.

As Tanya Sharma said, since the two batteries are not actually in series, you can't simply subtract their voltages.