[1 0 -22 -3-7 5 2-2 -4 3 1-21.0 1R =0 01-21.-10 22 1-5 -7 6 2 -7-410 00 0You may use, without justification, the fact that Ris the reduced echelon form of A.Determine the set of all solutions to the linear system2x1 – 3r2 – 7r3 + 5x4 + 2x5 = -221 – 2x2 – 4r3+3r4 + x5 = -22x1 – 4x3 + 2x4 + x5 = 3xi – 5x2 – 7x3 + 6x4 + 2x5 = -7and express this solution set in parametric vector form.

Answered: 2 -3 -7 5 2 -2] [1 0 -2 1 1 -4 3 1 -4 2…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

[1 0 -2
2 -3
-7 5 2
-2 -4 3 1
-2
1.
0 1
R =
0 0
1
-2
1.
-1
0 2
2 1
-5 -7 6 2 -7
-4
1
0 0
0 0
You may use, without justification, the fact that Ris the reduced echelon form of A.
Determine the set of all solutions to the linear system
2x1 – 3r2 – 7r3 + 5x4 + 2x5 = -2
21 – 2x2 – 4r3+3r4 + x5 = -2
2x1 – 4x3 + 2x4 + x5 = 3
xi – 5x2 – 7x3 + 6x4 + 2x5 = -7
and express this solution set in parametric vector form.

Expert Solution

Step 1

Given system of equation

$\begin{matrix} 2 x_{1} - 3 x_{2} - 7 x_{3} + 5 x_{4} + 2 x_{5} = - 2 \\ x_{1} - 2 x_{2} - 4 x_{3} + 3 x_{4} + x_{5} = - 2 \\ 2 x_{1} - 4 x_{3} + 2 x_{4} + x_{5} = 3 \\ x_{1} - 5 x_{2} - 7 x_{3} + 6 x_{4} + 2 x_{5} = - 7 \end{matrix}\} ....... (1)$

Write the matrix form $A x = b$

$\begin{matrix} A x = b \\ [\begin{matrix} 2 & - 3 & - 7 & 5 & 2 \\ 1 & - 2 & - 4 & 3 & 1 \\ 2 & 0 & - 4 & 2 & 1 \\ 1 & - 5 & - 7 & 6 & 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}] = [\begin{matrix} - 2 \\ - 2 \\ 3 \\ - 7 \end{matrix}] \end{matrix}$

Write the augmented matrix $(A | b)$

$(A | b) = [\begin{matrix} 2 & - 3 & - 7 & 5 & 2 & - 2 \\ 1 & - 2 & - 4 & 3 & 1 & - 2 \\ 2 & 0 & - 4 & 2 & 1 & 3 \\ 1 & - 5 & - 7 & 6 & 2 & - 7 \end{matrix}]$

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