Question

Test the consistency 2x-3y+7z=5,3x+y-3z=13,2x+19y-47z=32

Answer 1

hope it helps............

Answer 2

It is consistent and rank is 2.

Given

To test the given consistency.

                             2x - 3y + 7z = 5

                             3x + y - 3z = 13

                             2x + 19y - 47z = 32

To find consistency :

                   Δ = $\left[\begin{array}{ccc}2&-3&7\\3&1&-3\\2&19&-47\end{array}\right]$

                       = 2 ( -47+57 ) + 3 ( -141+6 ) + 7 ( 57-2 )          

                       = 2 ( 10 ) + 3 ( -135 ) + 7 ( 55 )

                       = 0.

The given equation is consistent.

To find rank :

Arranging x, y, z and constant terms,

                            $\left[\begin{array}{ccc}2&-3&7\\3&1&-3\\2&19&-47\end{array}\right] \left[\begin{array}{ccc}5\\13\\32\end{array}\right] = \left[\begin{array}{ccc}A&B\end{array}\right]$

Take difference of first row and second row,

                                   $R_{1}$ ⇒ $R_{1}-R_{2}$

                             $\left[\begin{array}{ccc}-1&-4&10\\3&1&-3\\2&19&-47\end{array}\right] \left[\begin{array}{ccc}-8\\13\\32\end{array}\right]$

                                    $R_{2}$ ⇒ $R_{2}+ 3R_{1}$

                             $\left[\begin{array}{ccc}-1&-4&10\\0&-11&27\\2&19&-47\end{array}\right] \left[\begin{array}{ccc}-8\\-11\\32\end{array}\right]$

                                     $R_{3}$ ⇒ $R_{3} + 2R_{1}$

                              $\left[\begin{array}{ccc}-1&-4&10\\0&-11&27\\0&11&-27\end{array}\right] \left[\begin{array}{ccc}-8\\-11\\16\end{array}\right]$

                                      $R_{3}$ ⇒ $R_{3} + R_{2}$

                               $\left[\begin{array}{ccc}-1&-4&10\\0&-11&27\\0&0&0\end{array}\right] \left[\begin{array}{ccc}-8\\-11\\5\end{array}\right]$

It is consistent and has rank = 2.

To learn more...

1. brainly.in/question/5545224

2. brainly.in/question/30094