Even more, what do you want to do? You have two functions from R to R^2, f(t)= (5t+ 1, -2t+ 1) and g(t)= (-15t- 2, 6t+ 2). (Those are not "f(x,y)" and "g(x,y)".) What do you want to do with them? What is your question?
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?