I would like to go back to your original demand.

On the first line, there was a title : "The problem". The second line is : "Balance the chemical equation using oxidation numbers". The third line is : "What I have tried". The fourth line is the equation between $\ce{Fe^{2+}}$ and $\ce{Cr2O7^{2-}}$, which is correct, although no stoichiometric coefficients are indicated. Up to here all is right. Nothing to object. "The main problem here is to define these coefficients. This is where you made a mistake.

The next two lines start with "Electron transition", follow by the dichromate reduction. Here you correctly find that $6$ electrons are needed when reducing $1$ $\ce{Cr2O7^{2-}}$ ion to $2$ $\ce{Cr^{3+}}$, although the formula of the chromium ions has a $+3$ charge, which was forgotten. Well. This typo has no consequence.

What is more serious is the following line, where you correctly state that the oxidation of $\ce{Fe^{2+}}$ to $\ce{Fe^{3+}}$ produces $1$ electron. $1$ electron par iron ion is correct. But just after this equation you multiply this equation by $3$. Strange decision. Why by $3$ ? There is no reason for multiplying by $3$ the equation $$\ce{Fe^{2+} -> Fe^{3+} + e-}$$ If you multiply this equation by $3$, you produce $3$ electrons. And these $3$ electrons cannot do anything ! To reduce one $\ce{Cr2O7^{2-}}$ ion, $3$ electrons are not useful. Instead of $3$, the ion $\ce{Cr2O7^{2-}}$ requires $6$ electrons. Six ! Not three ! The final equation must bring $6$ electrons ; so it must contain $6$ $\ce{Fe^{2+}}$ ions on the left-hand side.

This is the origin of your mistake. Once you have understood the origin this number $6$, I am sure you will find the rest of the problem, and obtain the final equation.

I would like to go back to your original demand.

On the first line, there was a title : "The problem". The second line is : "Balance the chemical equation using oxidation numbers". The third line is : "What I have tried". The fourth line is the equation between $\ce{Fe^{2+}}$ and $\ce{Cr2O7^{2-}}$, which is correct, although no stoichiometric coefficients are indicated. Up to here all is right. Nothing to object. The main problem here is to define these coefficients. This is where you made a mistake.

The next two lines start with "Electron transition", follow by the dichromate reduction. Here you correctly find that $6$ electrons are needed when reducing $1$ $\ce{Cr2O7^{2-}}$ ion to $2$ $\ce{Cr^{3+}}$, although the formula of the chromium ions has a $+3$ charge, which was forgotten. Well. This typo has no consequence.

What is more serious is the following line, where you correctly state that the oxidation of $\ce{Fe^{2+}}$ to $\ce{Fe^{3+}}$ produces $1$ electron. $1$ electron par iron ion is correct. But just after this equation you multiply this equation by $3$. Strange decision. Why by $3$ ? There is no reason for multiplying by $3$ the equation $$\ce{Fe^{2+} -> Fe^{3+} + e-}$$ If you multiply this equation by $3$, you produce $3$ electrons. And these $3$ electrons cannot do anything ! To reduce one $\ce{Cr2O7^{2-}}$ ion, $3$ electrons are not useful. Instead of $3$, the ion $\ce{Cr2O7^{2-}}$ requires $6$ electrons. Six ! Not three ! The final equation must bring $6$ electrons ; so it must contain $6$ $\ce{Fe^{2+}}$ ions on the left-hand side.

This is the origin of your mistake. Once you have understood the origin this number $6$, I am sure you will find the rest of the problem, and obtain the final equation.