Problem
ALG-B2-M08-P010 Representation of product 1
#10
★★★★☆ Level 4 of 5
Let \(a,b,c>0\), \(abc=1\). Show that one can choose \(x,y,z>0\) such that \(a=x/y\), \(b=y/z\), \(c=z/x\).
Hint. It is enough to take \(z=1\), then choose \(y\) and \(x\).
Set \(z=1\), \(y=b\), \(x=ab\). Then \(x/y=a\), \(y/z=b\), and \(z/x=1/(ab)=c\), since \(abc=1\).
Teaching goal: recognize the appropriate substitution and check the domain.