Symmetricality

Here is a set of five equations:

$$b+c+d+e=4\\
a+c+d+e=5\\
a+b+d+e=1\\
a+b+c+e=2\\
a+b+c+d=0$$

What do you notice when you add the five equations?

Can you now find the values of $a, b, c, d$ and $e$?

Here is a different set of equations:

$$xy = 1\\
yz = 4\\
zx = 9$$

What do you notice when you multiply the three equations given above? 

Can you now find the values of $x, y$ and $z$? 
Is there more than one possible set of values?

Here is a third set of equations:

$$ab = 1\\
bc = 2\\
cd = 3\\
de = 4\\
ea = 6$$

Can you find all the sets of values ${a, b, c, d, e}$ that satisfy these equations?
 

Extension

You may like to have a go at Overturning Fracsum.

Can you create your own set of symmetrical equations?