Question

A school wants to allocate students into three clubs: Sports, Music, and Drama, under the following conditions:
- The number of students in the Sports club should be equal to the sum of the number of students in the Music and Drama clubs.
- The number of students in the Music club should be 20 more than half the number of students in the Sports club.
- The total number of students to be allocated in all three clubs is 10.
Find the number of students allocated to different clubs, using the matrix method.}

Hint:

Matrix methods can be used to solve systems of linear equations by representing the system as a matrix equation and solving using matrix operations.

Answer 1

Let the number of students in the Sports, Music, and Drama clubs be \( x \), \( y \), and \( z \), respectively. The given conditions are: \[x = y + z, \quad y = \frac{x}{2} + 20, \quad x + y + z = 10.\] This system can be rewritten as: \[\begin{aligned}x - y - z &= 0, \\ y - \frac{x}{2} &= 20, \\ x + y + z &= 10.\end{aligned}\] To solve this system using matrix methods, represent it as a matrix equation: \[\begin{pmatrix}1 & -1 & -1 \\ - \frac{1}{2} & 1 & 0 \\ 1 & 1 & 1\end{pmatrix}\begin{pmatrix}x \\ y \\ z\end{pmatrix}=\begin{pmatrix}0 \\ 20 \\ 10\end{pmatrix}.\] Applying matrix operations to solve for \( x \), \( y \), and \( z \) yields: \[x = 60, \quad y = 50, \quad z = 70.\] Therefore, the number of students in the Sports, Music, and Drama clubs are 60, 50, and 70, respectively.