Manifold Problems
We have a subset X, which is contained in R^4 (i.e., it is contained in the reals in 4 dimensions).
(a) We must prove that the following two equations represent a manifold in the neighborhood of the point a = (1,0,1,0): (x_1)^2+(x_2)^2(x_3)^2(x_4)^2=0 and x_1+2x_2+3x_3+4x_4=4. (b) Also we must find a tangent space to X at a. (c) We must find a pair of variables that the equations above do not express as functions of the other two. (d) We must determine whether the enter set X is a manifold and prove the conclusion. How do you do this problem? Thanks. 
Use the definition of a manifold (or a theorem thereof). For example, a theorem dealing with level sets.

Can you elaborate, especially on level sets?

If g:R^{n} > R^{m}, you may have a theorem that states the conditions necessary for g^{1}(0) to be an (nm)dim. manifold. Such a set is called a level set of g.
For example, if g:R^{3} > R is the function g(x) = x  1, then g^{1}(0) is the 2sphere. The conditions in my book are that the domain of g be open, and that g be differentiable with rank m wherever g(x)=0. 
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