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 conditions are given as: \[ x = y + z, \quad y = \frac{x}{2} + 20, \quad x + y + z = 10. \] This system of equations can be written as: \[ \begin{aligned} x - y - z &= 0,
y - \frac{x}{2} - 20 &= 0,
x + y + z &= 10. \end{aligned} \] This system of equations can be solved using matrix methods. Write the system as a matrix equation: \[ \begin{pmatrix} 1 & -1 & -1
- \frac{1}{2} & 1 & 0
1 & 1 & 1 \end{pmatrix} \begin{pmatrix} x
y
z \end{pmatrix} = \begin{pmatrix} 0
20
10 \end{pmatrix}. \] Now, use matrix operations to solve for \( x \), \( y \), and \( z \), which will give the number of students in each club. The solution gives: \[ x = 60, \quad y = 50, \quad z = 70. \] Thus, the number of students in the Sports, Music, and Drama clubs are 60, 50, and 70, respectively.