Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$

For the given system of equation we have $x_{2}, x_{3}, x_{5}$ are free variables and $x_{1} = 134 - 2 x_{2} - 3 x_{3} - 5 x_{5}, x_{4} = 7 - x_{5}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Augmented matrix

Solving system of equations using gauss Jordan’s elimination method.

First of all we will make the augmented matrix to the corresponding system of equations.

$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$

Corresponding augmented matrix

$[\begin{matrix} 3 & 6 & 9 & 5 & 25 \\ 7 & 14 & 21 & 9 & 53 \\ - 4 & - 8 & - 12 & 5 & - 10 \end{matrix} \begin{matrix} 53 \\ 105 \\ 11 \end{matrix}]$

Step 2: Solving the Augmented matrix

$[\begin{matrix} 3 & 6 & 9 & 5 & 25 \\ 7 & 14 & 21 & 9 & 53 \\ - 4 & - 8 & - 12 & 5 & - 10 \end{matrix} \begin{matrix} 53 \\ 105 \\ 11 \end{matrix}] R_{1} \leftrightarrow R_{3} R_{1} \to (- 1) R_{1} - R_{3} ~ [\begin{matrix} 1 & 2 & 3 & - 10 & - 15 & - 64 \\ 7 & 14 & 21 & 9 & 53 & 105 \\ 3 & 6 & 9 & 5 & 25 & 53 \end{matrix}]$

$R_{2} \to R_{2} - 7 R_{1} R_{3} \to R_{3} - 3 R_{1} [\begin{matrix} 1 & 2 & 3 & - 10 & - 15 \\ 0 & 0 & 0 & 79 & 158 \\ 0 & 0 & 0 & 35 & 70 \end{matrix} \begin{matrix} - 64 \\ 553 \\ 245 \end{matrix}] R_{2} \to \frac{1}{79} R_{2} R_{1} \to \frac{1}{35} R_{3} [\begin{matrix} 1 & 2 & 3 & - 10 & - 15 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 & 2 \end{matrix} \begin{matrix} - 64 \\ 7 \\ 7 \end{matrix}] R_{1} \to R_{1} + 10 R_{2} R_{3} \to R_{3} - R_{2} [\begin{matrix} 1 & 2 & 3 & 0 & 5 & 134 \\ 0 & 0 & 0 & 1 & 2 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

Step 3: Value of variables

Corresponding system of equations will be

$|\begin{matrix} x_{1} + 2 x_{2} + 3 x_{3} + 5 x_{5} = 134 \\ x_{4} + x_{5} = 7 \end{matrix}| = |\begin{matrix} x_{1} = 134 - 2 x_{2} - 3 x_{3} - 5 x_{5} \\ x_{4} = 7 - x_{5} \end{matrix}|$

Here, $x_{2}, x_{3}, x_{5}$ are free variables.

Let the value of $x_{2}, x_{3}, x_{5} = 1$ , $\Rightarrow x_{1} = 124, x_{4} = 6$

Step 4: Final answer

For the given system of equation we have $x_{2}, x_{3}, x_{5}$ are free variables and $x_{1} = 134 - 2 x_{2} - 3 x_{3} - 5 x_{5}, x_{4} = 7 - x_{5}$ .

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Linear Algebra With Applications

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Short Answer

Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$

Step by Step Solution

Step 1: Augmented matrix

Step 2: Solving the Augmented matrix

Step 3: Value of variables

Step 4: Final answer

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Linear Algebra With Applications

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Short Answer

Solve the linear system of equations. You may use technology.|3x_1+6x_2+9x_3+5x_4+25x_5&=537x_1+14x_2+21x_3+9x_4+53x_5&=105-4x_1-8x_2-12x_3+5x_4-10x_5&=11|

Step by Step Solution

Step 1: Augmented matrix

Step 2: Solving the Augmented matrix

Step 3: Value of variables

Step 4: Final answer

Most popular questions for Math Textbooks

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Pure Maths

Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$