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Q14E

Expert-verifiedFound in: Page 1

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.

\({x_5} = 0\).

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.

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

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 \({x_3},{x_4}\).

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.

\({x_5} = 0\).

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