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Q66E

Expert-verifiedFound in: Page 23

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**At the beginning of a semester,${\mathbf{55}}$ ** **students have signed up for Linear Algebra; the course is offered in two sections that are taught at different times. Because of scheduling conflicts and personal preferences, ** ${\mathbf{20}}{\mathbf{\%}}$**of the students in Section${\mathit{A}}$ ** **switch to ${\mathit{B}}$Section** **in the first few weeks of class, while${\mathbf{30}}{\mathbf{\%}}$ ** **of the students in Section ${\mathit{B}}$** **switch to ${\mathit{A}}$** **, resulting in a net loss of $\mathbf{4}$** **students for Section${\mathit{B}}$ ** **. How large were the two sections at the beginning of the semester? No students dropped Linear Algebra (why would they?) or joined the course late.**

** **

At the beginning of the semester,$25$ students were in section A and students were in section B.

**The volume of sections A and B can be found using the augmented matrix. **

The total number of students is, $A+B=55\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{......}\left(1\right)$

A loss of 4 students occurred for Section B when$20\%$ of the students in Section A switched to Section B in the first few weeks of class and$30\%$ of the students in Section B switched to A.

$0.2A-0.3B=-4\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{......}\left(2\right)$

Consider the equations (1) and (2). The system of equations is,

$\left|\begin{array}{l}A+B=55\\ 0.2A-0.3B=-4\end{array}\right|$

Represent the above equations in terms of a matrix.

$\left|\begin{array}{ccc}1& 1& 55\\ 0.2& -0.3& -4\end{array}\right|$

** **

Consider the obtained matrix, now reduce it to row echelon form.

$\mathbf{rref}\left|\begin{array}{ccc}1& 1& 55\\ 0.2& -0.3& -4\end{array}\right|\to \left|\begin{array}{ccc}1& 0& 25\\ 0& 1& 30\end{array}\right|$

The values are, $A=25,B=30$.

Hence, at the beginning of the semester, $25$students were in section A and$30$ students were in section B.

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