Answer:
Gina rent 3 dramas, 5 comedies, and 3 documentaries.
Step-by-step explanation:
Let Gina rented x movies of dramas , y movies of comedies and z movies of documentaries then , Gina rented total 11 movies.
\(\Rightarrow x+y+z=11\) .........(1)
Also given Sam rented twice as many dramas, three times as many comedies, and twice as many documentaries Gina, thus he rented 2x movies of dramas , 3y movies of comedies and 2z movies of documentaries. also, he rented a total of 27 movies.
\(\Rightarrow 2x+3y+2z=27\) ............(2)
Also, Robby rented the same number of dramas, twice as many comedies, and twice as many documentaries as Gina, thus, he rented x movies of dramas , 2y movies of comedies and 2z movies of documentaries also, he rented a total of 19 movies.
\(\Rightarrow x+2y+2z=19\) ............(3)
Solving the three equation using matrix form,
\(\left[\begin{array}{ccc}1&1&1\\2&3&2\\1&2&2\end{array}\right] \left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}11\\27\\19\end{array}\right]\)
This, system is in form of AX= b,
Where, \(A=\left[\begin{array}{ccc}1&1&1\\2&3&2\\1&2&2\end{array}\right]\) , \(X=\left[\begin{array}{c}x\\y\\z\end{array}\right]\) , \(b=\left[\begin{array}{c}11\\27\\19\end{array}\right]\)
Pre-mutiply by A inverse both sides,
\(X=A^{-1}b\) ............(P)
First finding inverse,
\(\mathrm{Augment\:with\:a}\:3x3\:\mathrm{identity\:matrix}\)
\(=\begin{bmatrix}1&1&1&\mid \:&1&0&0\\ 2&3&2&\mid \:&0&1&0\\ 1&2&2&\mid \:&0&0&1\end{bmatrix}\)
\(\mathrm{Swap\:matrix\:rows:}\:R_1\:\leftrightarrow \:R_2\)
\(=\begin{bmatrix}2&3&2&\mid \:&0&1&0\\ 1&1&1&\mid \:&1&0&0\\ 1&2&2&\mid \:&0&0&1\end{bmatrix}\)
\(\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_2\:\mathrm{\:by\:performing}\:R_2\:\leftarrow \:R_2-\frac{1}{2}\cdot \:R_1\)
\(=\begin{bmatrix}2&3&2&\mid \:&0&1&0\\ 0&-\frac{1}{2}&0&\mid \:&1&-\frac{1}{2}&0\\ 1&2&2&\mid \:&0&0&1\end{bmatrix}\)
\(\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_3\:\mathrm{\:by\:performing}\:R_3\:\leftarrow \:R_3-\frac{1}{2}\cdot \:R_1\)
\(=\begin{bmatrix}2&3&2&\mid \:&0&1&0\\ 0&-\frac{1}{2}&0&\mid \:&1&-\frac{1}{2}&0\\ 0&\frac{1}{2}&1&\mid \:&0&-\frac{1}{2}&1\end{bmatrix}\)
\(\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_3\:\mathrm{\:by\:performing}\:R_3\:\leftarrow \:R_3+1\cdot \:R_2\)
\(=\begin{bmatrix}2&3&2&\mid \:&0&1&0\\ 0&-\frac{1}{2}&0&\mid \:&1&-\frac{1}{2}&0\\ 0&0&1&\mid \:&1&-1&1\end{bmatrix}\)
\(\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_1\:\mathrm{\:by\:performing}\:R_1\:\leftarrow \:R_1-2\cdot \:R_3\)
\(=\begin{bmatrix}2&3&0&\mid \:&-2&3&-2\\ 0&-\frac{1}{2}&0&\mid \:&1&-\frac{1}{2}&0\\ 0&0&1&\mid \:&1&-1&1\end{bmatrix}\)
\(\mathrm{Multiply\:matrix\:row\:by\:constant:}\:R_2\:\leftarrow \:-2\cdot \:R_2\)
\(=\begin{bmatrix}2&3&0&\mid \:&-2&3&-2\\ 0&1&0&\mid \:&-2&1&0\\ 0&0&1&\mid \:&1&-1&1\end{bmatrix}\)
\(\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_1\:\mathrm{\:by\:performing}\:R_1\:\leftarrow \:R_1-3\cdot \:R_2\)
\(=\begin{bmatrix}2&0&0&\mid \:&4&0&-2\\ 0&1&0&\mid \:&-2&1&0\\ 0&0&1&\mid \:&1&-1&1\end{bmatrix}\)
\(\mathrm{Multiply\:matrix\:row\:by\:constant:}\:R_1\:\leftarrow \frac{1}{2}\cdot \:R_1\)
\(=\begin{bmatrix}1&0&0&\mid \:&2&0&-1\\ 0&1&0&\mid \:&-2&1&0\\ 0&0&1&\mid \:&1&-1&1\end{bmatrix}\)
Thus, \(A^{-1}=\begin{pmatrix}2&0&-1\\ -2&1&0\\ 1&-1&1\end{pmatrix}\)
Put values in equation (P),
\(X=A^{-1}b\)
\(\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c,c,c}2&0&-1\\ -2&1&0\\ 1&-1&1\end{array}\right]\left[\begin{array}{c}11\\27\\19\end{array}\right]\)
\(\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c,c,c}2\cdot \:11+0\cdot \:27+\left(-1\right)\cdot \:19\\ \left(-2\right)\cdot \:11+1\cdot \:27+0\cdot \:19\\ 1\cdot \:11+\left(-1\right)\cdot \:27+1\cdot \:19\end{array}\right]\)
\(\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}3\\5\\3\end{array}\right]\)
Thus, Gina rent 3 dramas, 5 comedies, and 3 documentaries.