Problem
ALG-B1-M01-P012 A System Without Guessing
#12
★★★☆☆ Level 3 of 5
Find all real pairs \(x,y\) such that \(x+y=4\), \(x^3+y^3=28\).
Express \(x^3+y^3\) through \(s=x+y\) and \(p=xy\).
Let \(s=x+y=4\), \(p=xy\). Then \(x^3+y^3=s^3-3ps=64-12p\).
Since \(64-12p=28\), we get \(p=3\). Thus \(x,y\) are roots of \(t^2-4t+3=0\), so \(t=1\) or \(t=3\). The pairs are \((1,3)\), \((3,1)\).
The student should recover sum and product, not guess the pair.