Squaring x + y + z = 0 gives x2 + y2 + z2 + 2(xy + yz + zx) = 0. Since xy + yz + zx = 0, it simplifies to x2 + y2 + z2 = 0. Since squares are non-negative, x, y, z must be 0.
Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero.
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We are given two equations: x + y + z = 0 and xy + yz + zx = 0. Using the algebraic identity, we know that (x + y + z) squared equals x2 + y2 + z2 + 2(xy + yz + zx). Substituting our given values into this identity results in 0 squared equals x2 + y2 + z2 + 2(0), which simplifies directly to x2 + y2 + z2 = 0. Since the square of any real rational number is always non-negative, their sum can only equal zero if x, y, and z are all simultaneously zero.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3 The world of numbers (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-3/