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### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Find the general solutions of the systems whose augmented matrices are given as 14. $$\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&{ - 6}&0&{ - 5}\\0&1&{ - 6}&{ - 3}&0&2\\0&0&0&0&1&0\\0&0&0&0&0&0\end{array}} \right]$$.

The general solution of the system is

$${x_1} = - 5 + 2{x_2} + 5{x_3} + 6{x_4}$$

$${x_2} = 2 + 6{x_3} + 3{x_4}$$

$${x_3}$$ is free.

$${x_4}$$is free.

$${x_5} = 0$$.

See the step by step solution

## Step 1: Convert the given matrix into the equation

To obtain the general solution of the system, you have to convert the augmented matrix into the system of equations.

Write the given matrix into the equation notation.

\begin{aligned}{c}{x_1} - 2{x_2} - 5{x_3} - 6{x_4} = - 5\\{x_2} - 6{x_3} - 3{x_4} = 2\\{x_5} = 0\end{aligned}

## Step 2: Determine the basic variable and free variable

The variables corresponding to the pivot columns in the matrix are called basic variables.

The other variables are called free variables.

The basic variables of the given matrix are $${x_1},{x_2},{x_5}$$. The free variables are $${x_3},{x_4}$$.

## Step 3: Write the general solution of the system

Thus, the general solution of the system is

\begin{aligned}{c}{x_1} = - 5 + 2{x_2} + 5{x_3} + 6{x_4}\\{x_2} = 2 + 6{x_3} + 3{x_4}\end{aligned}

$${x_3}$$ is a free variable.

$${x_4}$$ is a free variable.

$${x_5} = 0$$.