Problem
ALG-B1-M06-P014 Fractions with zero sum
#14
★★★☆☆ Level 3 of 5
Let \(x,y,z\neq 0\), \(x+y+z=0\), and assume none of the denominators below is zero. Prove that \[\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=-3.\]
Replace \(y+z\), \(z+x\), \(x+y\) by the negatives of the remaining variables.
Since \(x+y+z=0\), we have \(y+z=-x\), \(z+x=-y\), \(x+y=-z\). Therefore the sum is \(\frac{x}{-x}+\frac{y}{-y}+\frac{z}{-z}=-1-1-1=-3\).
The point is that the student should not use a common denominator unnecessarily.