How to determine flatness

Question

Let $k$ be a field and define $R=k[x,y]/(xy)$ and an $R$-Algebra $A=k[x]\times k[y]=R/(y)\times R/(x)$. I have to answer two questions

1. Is $k[x]=R/(y)$ flat over $A$
2. Is $A$ flat over $R$

I don't necessarily need the answer but I don't know where to start with both these questions. I do know the definition of flatness but is there a way to not use exact sequences with tensor products? Or is that the only way? So any hint is appreciated.

Edit: As suggested by someone: We closely follow the book of Atiyah and Macdonald introduction to commutative algebra and we are in chapter 4.

Answer 1

Since you only wanted a hint:

1. Lemma: Projective modules are flat.

2. Lemma: If $I$ is an ideal of a commutative ring $R$ and $R \to R/I$ is flat, then $I = I^2$.