In general, when working with symbolic algorithms, you should make sure your input is symbolic. For example, you want to use symbolic numbers. Number with a decimal point in them such as "1." and "2." are not symbolic numbers. The are floating point numbers. So first, I would remove the decimal points in the code above.
The second issue is that Solve doesn't know whether these variables are supposed to be Real numbers or Complex numbers. See the documentation for Solve. There is a way to specify the domain over which you want your solutions. In many cases people want to evaluate:
Solve[{eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, eq10}, {Ly, Lq, LA, gA, gB, r, g1, g2, g3, p}, Reals]
I don't know if this equation has a solution without working it out myself. It looks like your equations are multinomials. In that case you might be able to use GrobnerBasis to solve or at least reduce the complexity of set of equations. Using GobernerBasis is beyond the scope of what I can talk about here, but there are examples in the documentation and resources online which explain what it is and how it works.
