Solving a system of equations with sparse data
If I understand you correctly, this is the case of multiple linear regression with sparse data (sparse regression). Assuming that, I hope you will find the following resources useful.

1) NCSU lecture slides on sparse regression with overview of algorithms, notes, formulas, graphics and references to literature: http://www.stat.ncsu.edu/people/zhou/courses/st810/notes/lect23sparse.pdf

2) R ecosystem offers many packages, useful for sparse regression analysis, including:


Matrix (http://cran.r-project.org/web/packages/Matrix)
SparseM (http://cran.r-project.org/web/packages/SparseM)
MatrixModels (http://cran.r-project.org/web/packages/MatrixModels)
glmnet (http://cran.r-project.org/web/packages/glmnet)
flare (http://cran.r-project.org/web/packages/flare)


3) A blog post with an example of sparse regression solution, based on SparseM: http://aleph-nought.blogspot.com/2012/03/multiple-linear-regression-with-sparse.html

4) A blog post on using sparse matrices in R, which includes a primer on using glmnet: http://www.johnmyleswhite.com/notebook/2011/10/31/using-sparse-matrices-in-r

5) More examples and some discussion on the topic can be found on StackOverflow: https://stackoverflow.com/questions/3169371/large-scale-regression-in-r-with-a-sparse-feature-matrix

UPDATE (based on your comment):

If you're trying to solve an LP problem with constraints, you may find this theoretical paper useful: http://web.stanford.edu/group/SOL/papers/gmsw84.pdf.

Also, check R package limSolve: http://cran.r-project.org/web/packages/limSolve. And, in general, check packages in CRAN Task View "Optimization and Mathematical Programming": http://cran.r-project.org/web/views/Optimization.html.

Finally, check the book "Using R for Numerical Analysis in Science and Engineering" (by Victor A. Bloomfield). It has a section on solving systems of equations, represented by sparse matrices (section 5.7, pages 99-104), which includes examples, based on some of the above-mentioned packages: http://books.google.com/books?id=9ph_AwAAQBAJ&amp;pg=PA99&amp;lpg=PA99&amp;dq=r+limsolve+sparse+matrix&amp;source=bl&amp;ots=PHDE8nXljQ&amp;sig=sPi4n5Wk0M02ywkubq7R7KD_b04&amp;hl=en&amp;sa=X&amp;ei=FZjiU-ioIcjmsATGkYDAAg&amp;ved=0CDUQ6AEwAw#v=onepage&amp;q=r%20limsolve%20sparse%20matrix&amp;f=false.