Finding all the sets of three real numbers that satisfy specific equations I am having trouble finding a set of three numbers of real numbers $(x, y, z)$ satisfying $x + y + z = xy + xz + yz = 3$. I have tried factoring the equations around but I'm not having any luck and I don't think I'm going on the right path.

Given
$$a+b+c=3\tag1$$
$$ab+bc+ca=3\tag2$$
Hint:
$$(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)\tag3$$

Spoiler
From $(3)$ it can easily be shown that
$$a^2+b^2+c^2 = 3$$
Now
$$(a+b+c)^2 = 3^2$$
$$(a+b+c)^2 + a^2+b^2+c^2 - 4(ab+bc+ca) = 3^2 + 3 - 4*3$$
$$a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2 = 0$$
$$(a-b)^2+(b-c)^2+(c-a)^2=0$$
As $a,b,c$ are real numbers and squares of real numbers cannot be negative so
$a-b=b-c=c-a=0;a=b=c$
Replacing in $(3)$ we get
$$a+a+a=b+b+b=c+c+c=3$$
$$a=b=c=1$$
