an algebra problem lol

Algebra Level 2

Are there any positive integers $x$ and $y$ that satisfy the equation: $x^y +y^2= xy$ ?

(If yes, please tell me the solutions you can find in the discussions; if no, please tell me why in the discussions.)

Yes No

2 solutions

Chris Lewis
Jan 20, 2020

By the AM-GM inequality, $x^y+y^2 \ge 2\sqrt{x^y y^2} = xy \times 2\sqrt{x^{y-2}}$ . So we need $2\sqrt{x^{y-2}} \le 1$ .

If $y>1$ , the left-hand side is certainly greater than $1$ , so this is only possible if $y=1$ .

However, the equation then becomes $x+1=x$ , which clearly has no solutions.

Callie Ferguson
Jan 18, 2020

If you graph it (I used geogebra.com), you can pretty much tell from the graph that there aren't any integers that satisfy the equation.