Question

A furniture workshop produces three types of furniture: chairs, tables, and beds each day. On a particular day, the total number of furniture pieces produced is 45. It was also found that the production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using the matrix method.

Hint:

Use the matrix method to solve systems of linear equations. First, convert the system to matrix form, calculate the determinant, and find the inverse of the coefficient matrix to solve for the variables.

Answer 1

Let \( x \), \( y \), and \( z \) represent the quantities of chairs, tables, and beds produced, respectively. The problem is defined by the following system of equations: 1. \( x + y + z = 45 \) (total furniture production) 2. \( z = x + 8 \) (bed production exceeds chair production by 8) 3. \( x + z = 2y \) (combined production of beds and chairs equals twice the table production) This system can be expressed in matrix form: \[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 45 \\ 8 \\ 0 \end{bmatrix} \] To solve this system using matrix algebra, we first determine the inverse of the coefficient matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{bmatrix} \). The determinant of \( A \) is calculated as: \[ \text{det}(A) = 1 \cdot \begin{vmatrix} 0 & -1 \\ -2 & 1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 0 \\ 1 & -2 \end{vmatrix} \] \[ \text{det}(A) = 1 \cdot ((0 \cdot 1) - (-1 \cdot -2)) - 1 \cdot ((1 \cdot 1) - (-1 \cdot 1)) + 1 \cdot ((1 \cdot -2) - (1 \cdot 0)) \] \[ \text{det}(A) = 1 \cdot (-2) - 1 \cdot (2) + 1 \cdot (-2) = -2 - 2 - 2 = -6 \] After computing the inverse of \( A \), the system is solved to find the values of \( x \), \( y \), and \( z \). The solution yields: \( x = 17, \quad y = 14, \quad z = 18 \). Therefore, 17 chairs, 14 tables, and 18 beds were produced.