Short Answer

Rule of Simultaneous Solutions in Quality Control

A quality control technician is checking if the machine calibration settings x=−4x = -4, y=−3y = -3, and z=4z = 4 satisfy a system of three linear calibration constraints:

  1. x−y+z=2x - y + z = 2
  2. 2x−y−z=−6{}2x - y - z = -6
  3. 2x+2y+z=−3{}2x + 2y + z = -3

Upon substituting these values into the first constraint equation, x−y+z=2x - y + z = 2, the technician finds that the expression simplifies to the false statement 3=2{}3 = 2.

State the fundamental mathematical rule regarding systems of equations that allows the technician to immediately conclude that the ordered triple (-4, -3, 4) is not a solution to the entire system, without needing to test the remaining two equations.

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Updated 2026-09-18

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