**Problem 1**: Solve the linear system\[\begin{align*}x_1 + x_2 + x_3 + x_4 &= 1 \\3x_1 + x_2 + x_3 - x_4 &= 5 \\5x_1 - x_2 + x_3 - 5x_4 &= 3\end{align*}\]**Task**: Describe the solution set geometrically.

Answered: Solve the linear system X1 + x₂ + x3 +…

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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**Problem 1**: Solve the linear system

\[
\begin{align*}
x_1 + x_2 + x_3 + x_4 &= 1 \\
3x_1 + x_2 + x_3 - x_4 &= 5 \\
5x_1 - x_2 + x_3 - 5x_4 &= 3
\end{align*}
\]

**Task**: Describe the solution set geometrically.

Expert Solution

Step 1

The augmented matrix for given system of equations is

$[A B] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 3 & 1 & 1 & - 1 & 5 \\ 5 & - 1 & 1 & - 5 & 3 \end{matrix}]$

Apply the row operators $R_{2} \to R_{2} - 3 R_{1} a n d R_{3} \to R_{3} - 5 R_{1}$

$[A B] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & - 2 & - 2 & - 4 & 2 \\ 0 & - 6 & - 4 & - 10 & - 2 \end{matrix}]$

Apply $R_{3} \to R_{3} - 3 R_{2}$

$[A B] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & - 2 & - 2 & - 4 & 2 \\ 0 & 0 & 2 & 2 & - 8 \end{matrix}]$

Apply $- \frac{1}{2} R_{2} a n d \frac{1}{2} R_{3}$

$[A B] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 2 & - 1 \\ 0 & 0 & 1 & 1 & - 4 \end{matrix}]$

This is row reduced echelon (RREF) form of the matrix AB. Since number of equations(3) is less than to number of variables(4) so the system has infinite number of solutions.

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Solve the linear system X1 + x₂ + x3 + x4 = 1 3x₁ + x₂ + x3 x4 = 5 5x₁ - x2 + x3 - 5x4 = 3 Describe the solution set geometrically.