Find three positive integers x, y, z that satisfy the given conditions. The product is 27, and the sum is a minimum.
I'm lost on how I would solve my system of equations that I have set up.
$1=\lambda \cdot yz$
$1=\lambda \cdot xz$
$1=\lambda \cdot xy$
$xyz=27$
Are my system of equations correct?
Best Answer
Yes, the system is$$\left\{\begin{array}{l}1=\lambda yz\\1=\lambda xz\\1=\lambda xy\\xyz=27.\end{array}\right.$$It follows from any of the first $3$ equations that none of the numbers $x$, $y$, and $z$ can be $0$. So$$1=\frac11=\frac{\lambda yz}{\lambda xz}=\frac yx;$$that is, $y=x$. By the same argument, $z=x=y$. Can you take it from here?