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$$x+xy+y=223$$ $$x^2 y+x y^2=5460$$

I need to find the integer solutions to this equation. However, from the looks of it a simple substitution and solve will be difficult, so it seems that clever manipulations might be necessary. I noticed that $(x+xy+y)^2$ includes a $x^2 y + xy^2$ term (multiplied by 2) but playing around with that seemed fruitless. Further, I also tried completing the rectangle on $x + xy+ y$ to get $(x+1)(y+1)$ but that too led to a dead end.

Michael Hardy
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3 Answers3

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Let $x+y=s$ and $x\cdot y=p$. Hence your system is:\begin{cases}s+p=223\\s\cdot p=5460\end{cases}

Now solve this and go back to solve a similar one.

Jika
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Do you know how to find $a$ and $b$ if you are given the sum $a+b$ and product $ab$ of $a$ and $b$? Here, you can use that trick twice: you are given the sum and the product of $x+y$ and $xy$, from which you can determine $x+y$ and $xy$; after that you can determine $x$ and $y$ themselves.

user133281
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The following might help: $$ xy(x+xy+y) = 223(xy) \\ \implies x^2y + xy^2 + (xy)^2 = 223(xy) \\ \implies (xy)^2 - 223(xy)+5460=0. $$ Using the quadratic equation to solve for $xy$ you get solutions $xy = 28$ or $xy = 195$. From here, perhaps you can check factors of $28$ and $195$...

Tom
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