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 the number of chairs, tables, and beds produced be denoted by \( x \), \( y \), and \( z \) respectively. We are given the following system of equations: 1. \( x + y + z = 45 \) (total furniture pieces) 2. \( z = x + 8 \) (production of beds exceeds that of chairs by 8) 3. \( x + z = 2y \) (total production of beds and chairs together is twice the production of tables) We can rewrite these equations in matrix form as: \[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 45 \\ 8 \\ 0 \end{bmatrix} \] Now, we solve this system using matrix algebra. First, we calculate the inverse of the coefficient matrix. The coefficient matrix is: \[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{bmatrix} \] The determinant of \( A \) is: \[ \text{det}(A) = 1 \cdot \begin{vmatrix} 0 & -1 \\ -2 & 1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 0 \\ 1 & -2 \end{vmatrix} \] \[ \text{det}(A) = 1 \cdot ((0 \cdot 1) - (-1 \cdot -2)) - 1 \cdot ((1 \cdot 1) - (-1 \cdot 1)) + 1 \cdot ((1 \cdot -2) - (1 \cdot 0)) \] \[ \text{det}(A) = 1 \cdot (-2) - 1 \cdot (2) + 1 \cdot (-2) = -2 - 2 - 2 = -6 \] The inverse of \( A \) can now be computed, and the result will give us the values of \( x \), \( y \), and \( z \). Solving the system: \[ x = 17, \quad y = 14, \quad z = 18 \] Thus, the number of chairs produced is \( 17 \), the number of tables produced is \( 14 \), and the number of beds produced is \( 18 \).