Proving $(x + y, z - 1)$ and $(x - z^2, y + z) \subseteq k[x,y,z]$ as ideals are radical

Question

Having a hard time thinking of how to tackle this. It's part 1 of a much longer assignment. Hints before outright solutions would be much appreciated.

Answer 1

Here's some start-help on how to compute the quotient of the second ideal.

First observe that one of the generators is linear. This always means that we can eliminate one of the variables:

$$ k[x,y,z]/(x-z^2,y+z) \stackrel{y \mapsto -z}{\simeq} k[x,z]/(x-z^2) $$

Here we eliminated $y$, since in the quotient we have that $y \equiv -z$.

Do you see how you an also eliminate $x$?