Real algebraic solution Suppose a system of polynomial equations with rational coefficients has a real solution. Does necessarily there exists a real solution with algebraic coordinates? What about the simplest case of one polynomial equation in two variables? 
 A: The is an immediate application of the "real Nullstellensatz", in exactly the same way that one answers the analogous questions between $\overline{\mathbf{Q}}$ and $\mathbf{C}$ by using the usual Nullstellensatz (over $\overline{\mathbf{Q}}$!).  I trust that the OP knows this latter application of the usual Nullstellensatz. It is just a proof by contradiction.
Namely, suppose $F \hookrightarrow F'$ is an inclusion between real closed fields (such as $F' = \mathbf{R}$ and $F$ the subfield of algebraic numbers inside $\mathbf{R}$) and let $f_1, \dots, f_n$ be a finite collection of polynomials in variables $X_1, \dots, X_m$ over $F$.  Given that the $f_i$'s have a common zero in $F'$ we want to claim the same in $F$. Suppose to the contrary that the $f_i$'s have no common zero in $F$.  Then the "real Nullstellensatz" (for real closed fields) says that $-1 = \sum g_j f_j + \sum h_k^2$ for some $g_j, h_k \in F[X_1,\dots,X_m]$.  But this identity persists over $F'$, where it obstructs the existence of a common zero of the $f_j$'s over $F'$, a contradiction.
A: The statement also follows from Tarski's theorem (1951) that the first-order theory of real closed fields is quantifier-eliminable. Namely, this theorem implies that if a first-order formula holds in some real closed field, then it holds in all real closed fields. One can apply this result to the field of real numbers and the field of real algebraic numbers (both of which are real closed fields). In particular, if a system of polynomial equations with rational coefficients has a real solution, then it also has a real algebraic solution. 
In fact this proof gives a bit more: the real algebraic solutions are dense among the real solutions (the topology is the usual one coming from the Euclidean metric).
I know two more proofs, let me just give the key ideas. The first one is based on the fact (a consequence of the Tarski-Seidenberg principle) that the projection of a semialgebraic set in $\mathbb{R}^n$ to $\mathbb{R}^{n-1}$ (by forgetting the last coordinate) is semialgebraic. The second one is based on Theorem 3.1 in Chapter IX of Lang's algebra.
