Question

Write the system of linear equations in \( x \) and \( y \) formed from the given situation.

Answer 1

Step 1: Total amount donated is the same in both cases. - Case 1: When there are \( x - 8 \) children, each gets \( y + 10 \): \[ (x - 8)(y + 10) = xy. \] Simplify: \[ xy - 8y + 10x - 80 = xy \quad \Rightarrow \quad -8y + 10x = 80 \quad \Rightarrow \quad 10x - 8y = 80. \quad \cdots (1) \] - Case 2: When there are \( x + 16 \) children, each gets \( y - 10 \): \[ (x + 16)(y - 10) = xy. \] Simplify: \[ xy + 16y - 10x - 160 = xy \quad \Rightarrow \quad 16y - 10x = 160. \quad \cdots (2) \] Step 2: The system of linear equations is: \[ 10x - 8y = 80, \quad -10x + 16y = 160. \]

Answer 2

The system of linear equations: \[ 10x - 8y = 80, \quad -10x + 16y = 160. \]
Matrix form \( AX = B \): \[ A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
Thus, \[ AX = B \quad \text{is written as:} \quad \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]

Answer 3

Matrix \( A \): \[ A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}. \]
Step 1: Compute the determinant of \( A \): \[ \det(A) = (10)(16) - (-8)(-10) = 160 - 80 = 80. \]
Step 2: Compute the adjoint of \( A \): \[ \text{Adj}(A) = \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Step 3: Compute the inverse of \( A \): \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{Adj}(A) = \frac{1}{80} \cdot \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Simplify: \[ A^{-1} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix}. \]

Answer 4

Step 1: Use \( X = A^{-1}B \): \[ X = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix} \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
Step 2: Perform the matrix multiplication: \[ x = 0.2(80) + 0.1(160) = 16 + 16 = 32, \] \[ y = 0.125(80) + 0.125(160) = 10 + 20 = 30. \]
Final Answer: \[ x = 32, \quad y = 30. \]