Question

Write a system of equations to describe the situation below, solve using an augmented matrix.An event organizer is reserving rooms for two company-wide events. For the quarterly meeting this month, she reserved 1 ballroom, which can seat a total of 46 attendees. For safety training next month, she reserved 2 conference rooms and 3 ballrooms, which can seat 166 attendees. How many attendees can each room accommodate?Each conference room can accommodate ? attendees and every ballroom can accommodate ? attendees.

Answer 1

Conference room: 14 attendees

Ballroom: 46 attendees

Explanation:

We have the following:

x : Number of attendees the conference room can accommodate

y : Number of attendees the ballroom can accommodate

With the data from the statement, we can establish the following augmented matrix:

`\begin{pmatrix}0&1\\ \:2&3\end{pmatrix}\begin{pmatrix}x\\ \:y\end{pmatrix}=\begin{pmatrix}46\\ \:166\end{pmatrix}`

We solve the matrix as follows:

`\begin{gathered} \begin{pmatrix}0 & 1 \\ 2 & 3\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}0\cdot\:x+1\cdot\:y \\ 2x+3y\end{pmatrix} \\ \\ \begin{pmatrix}y\\ 2x+3y\end{pmatrix}=\begin{pmatrix}46\\ 166\end{pmatrix} \\ \\ \begin{bmatrix}y=46\\ 2x+3y=166\end{bmatrix} \\ \\ \begin{bmatrix}2x+3\cdot\:46=166\end{bmatrix}\rightarrow\begin{bmatrix}2x+138=166\end{bmatrix}\rightarrow\begin{bmatrix}2x=166-138\end{bmatrix}\rightarrow\begin{bmatrix}x=(28)/(2)\end{bmatrix}\rightarrow\begin{bmatrix}x=14\end{bmatrix} \\ \\ \begin{bmatrix}x=14 \\ y=46\end{bmatrix} \end{gathered}`

Therefore:

Each conference room can accommodate 14 attendees and every ballroom can accommodate 46 attendees.