Question: The sum of the cubes of two numbers is $2071$, while the sum of the two
numbers themselves is $19$. Find the two numbers.

I've been working hard to solve this problem and I need someone to tell me how to solve it, it perplexes me.

I know the answer is $7$ and $12$ and it might have something to do with $$a^3 + b^3 = (a + b)(a^2 – ab + b^2)$$ but how do they get to that answer?

Any help will be appreciated thank you.

I followed this Yahoo question up to where it said solve it then I searched like $4$ hours looking for a way to solve it, so I'm hoping someone can help me out.

![enter image description here](https://i.stack.imgur.com/SvjD3.png)

Answer: 
So you have $$x+y=19$$ and $$x^2-xy+y^2=109$$ now just substitute $y = 19-x$ into the second equation, to get $$x^2-x(19-x)+(19-x)^2=109$$ After simplifying, we obtain $$x^2 - 19x+ 84=0$$

Solving this equation, we will get $x=7$ or $12$. Now, by symmetry, the $2$ numbers will be $7$ and $12$.
Hint: Complete the square. 

$$x^2-xy+y^2 = (x+y)^2 -3xy$$