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Expert-verified Found in: Page 18 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss–Jordan elimination. Show all your work.$|\begin{array}{c}{x}_{1}-7{x}_{2}+{x}_{5}=3\\ {x}_{3}-2{x}_{5}=2\\ {x}_{4}+{x}_{5}=1\end{array}|$

For the given system of equation ${x}_{5},{x}_{5}.$are free variables and

${x}_{3}=2+2{x}_{5,}{x}_{4}=1-{x}_{5,}=3+7{x}_{2}-{x}_{5}$.

See the step by step solution

## Step 1: Write the augmented matrix

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

$\left|\begin{array}{c}{x}_{1}-7{x}_{2}+{x}_{5}=3\\ {x}_{3}-2{x}_{5}=2\\ {x}_{4}+{x}_{5}=1\end{array}\right|$

The corresponding augmented matrix is written as:

$\left[\begin{array}{ccccc}1& -7& 0& 0& 1\\ 0& 0& 1& 0& -2\\ 0& 0& 0& 1& 1\end{array}\overline{)\begin{array}{c}3\\ 2\\ 1\end{array}}\right]$

## Step 2: Solving the Augmented matrix

The above augmented matrix obtained above is already in reduced form, then the

System of equation to the corresponding above matrix will be

$\left|\begin{array}{c}{x}_{1}=3+7{x}_{2}-{x}_{5}\\ {x}_{3}=2+2{x}_{5}\\ {x}_{4}=1-{x}_{5}\end{array}\right|$

## Step 3: Finding free variables

According to system $\left|\begin{array}{c}{x}_{1}=3+7{x}_{2}-{x}_{5}\\ {x}_{3}=2+2{x}_{5}\\ {x}_{4}=1-{x}_{5}\end{array}\right|$, the variables ${\mathrm{x}}_{3},{\mathrm{x}}_{4},{\mathrm{x}}_{1}$ depend on ${\mathrm{x}}_{2},{\mathrm{x}}_{5}.\mathrm{So},{\mathrm{x}}_{2},{\mathrm{x}}_{5}.$are free variables.

Thus, for the given system of equation ${\mathrm{x}}_{2,}{\mathrm{x}}_{5}$ are free variables and ${\mathrm{x}}_{3}=2+2{\mathrm{x}}_{5},{\mathrm{x}}_{4}=1-{\mathrm{x}}_{5}{\mathrm{x}}_{1}=3+7{\mathrm{x}}_{2}-{\mathrm{x}}_{5}$. 