Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.1 Exercises - Page 10: 11

Answer

System has no solutions. It is inconsistent.

Work Step by Step

The procedure is shown with the matrix notation for simpler understanding. $x_{2} + 4x_{3} = -5 $ $x_{1} + 3x_{2} + 5x_{3} = -2$ $3x_{1} + 7x_{2} + 7x_{3} = 6$ This can be depicted in the augmented matrix notation as follows : $\begin{bmatrix} 0 & 1 & 4 & -5 \\ 1 & 3 & 5 & -2 \\ 3 & 7 & 7 & 6 \end{bmatrix}$ To obtain an $x_{1}$ in the first equation,interchange rows 1 and 2: $\begin{bmatrix} 1 & 3 & 5 & -2 \\ 0 & 1 & 4 & -5 \\ 3 & 7 & 7 & 6 \end{bmatrix}$ To eliminate the $3x_{1}$ term in the third equation, add $-3$ times row 1 to row 3: $\begin{bmatrix} 1 & 3 & 5 & -2 \\ 0 & 1 & 4 & -5 \\ 0 & -2 & -8 & 12 \end{bmatrix}$ Next we use the $x_{2}$ term in the second equation to eliminate the $-2x_{2}$ term from the third equation. Add $2$ times row 2 to row 3: $\begin{bmatrix} 1 & 3 & 5 & -2 \\ 0 & 1 & 4 & -5 \\ 0 & 0 & 0 & 2 \end{bmatrix}$ Now the augmented matrix is in a triangular form. For interpreting it, we go back to the equation notation: $x_{1} + 3x_{2} + 5x_{3} = -2$ $x_{2} + 4x_{3} = -5 $ $0 = 2$ The equation $0 = 2$ is a short form of $0x_{1} + 0x_{2} + 0x_{3} = 2$. This is a contradiction, and no values of $ x_{1}, x_{2} $, and $x_{3} $ can satisfy this equation. Since the final and original equations have the same solution set, we conclude that the given system of equations $x_{2} + 4x_{3} = -5 $ $x_{1} + 3x_{2} + 5x_{3} = -2$ $3x_{1} + 7x_{2} + 7x_{3} = 6$ has no solutions. It is inconsistent.
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