Question

If \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \] then the value of \[ \left(\frac{24}{x} + \frac{24}{y}\right) \] is:

• A. 7
• B. 6
• C. 8
• D. 18
• E.

Hint:

To solve such problems, equate corresponding elements of the matrices and use algebraic identities to simplify expressions efficiently.

Answer 1

The Correct Option is D

Given the matrix equation: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}. \] 1. By equating corresponding elements: * \( x + y = 6 \) * \( xy = 8 \) 2. The expression to evaluate is: \[ \frac{24}{x} + \frac{24}{y}. \] Using the identity \( \frac{a}{x} + \frac{a}{y} = a \cdot \frac{x + y}{xy} \), we simplify the expression: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{x + y}{xy}. \] 3. Substituting the known values \( x + y = 6 \) and \( xy = 8 \): \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{6}{8}. \] Simplifying further: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{3}{4} = 18. \] Therefore, the value of \( \frac{24}{x} + \frac{24}{y} \) is 18.