The post, General expression for $\hat \beta_1$ in multiple linear expression is asking for a closed form expression of part of the coefficient vector in a linear regression model. Nevertheless it has been closed as a duplicate of a question asking for a closed form solution form the entire coefficient vector.
To me it is not really clear that these two questions are the same. Actually to me it is pretty clear that they are not the same under one interpretation.
Overall the question is a little unclear, but the present closure of the question makes it more difficult to clarify what the OP was really interested in.
What I do not understand is how this comes to be a duplicate of the cited question.
(I am partly asking with the following discussion in mind Are we closing questions too fast?)