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Found in: Page 23

### Linear Algebra With Applications

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${\mathbit{A}}$ switch to ${\mathbit{B}}$Section in the first few weeks of class, while${\mathbf{30}}{\mathbf{%}}$ of the students in Section ${\mathbit{B}}$ switch to ${\mathbit{A}}$ , resulting in a net loss of $\mathbf{4}$ students for Section${\mathbit{B}}$ . 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.

See the step by step solution

## Step 1: Represent the number of students in sections A and B in terms of system of equations.

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

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

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.

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

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

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

Represent the above equations in terms of a matrix.

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

## Step 2: Perform the row operations on the matrix.

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

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

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.