System of linear equations

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I am looking at a particular form of a set of linear equations. On the attached picture the form is shown for the case of 8 linear equations. It should not be hard to see how the set would look for an arbitrary N. My question is: Can anyone see if this special set of equations can be solved in a closed form for arbitrary N. That is given that I have N linear equations with the form as indicated, I can immidiatly write down:
x1 = (f1,f2,f3...., g2,g3,g4...), x2 = (....) ....
 

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    Linear equation.png
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