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Exercise 1.2.2

Avg difficulty 3.0

Consider the system of equations

x + 2y + 3z + 4w = 4x + y + z + w = 2x + 2y + 2z + w = 2.\begin{array}{ccccccccc}x~{}&+~{}&2y~{}&+~{}&3z~{}&+~{}&4w~{}&=~{}&4\\ x~{}&+~{}&y~{}&+~{}&z~{}&+~{}&w~{}&=~{}&2\\ x~{}&+~{}&2y~{}&+~{}&2z~{}&+~{}&w~{}&=~{}&2.\end{array}

For each of the following tuples (x,y,z,w)(x,y,z,w) of real numbers, determine whether it is a solution of the first equation, second equation, and/or third equation. Which ones are solutions to the system of equations?

(a) (2,0,−2,2)(2,0,-2,2)  (b) (2,2,−2,0)(2,2,-2,0)  (c) (1,1,−1,1)(1,1,-1,1)  (d) (3,0,−1,1)(3,0,-1,1)  (e) (2,−2,2,0)(2,-2,2,0)

Selinger, Matrix Theory and Linear Algebra, Systems of linear equations, Algebraic view of systems of equations, Exercise 1.2.2

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.