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:
Show Hint
- 7
- 6
- 8
- 18
The Correct Option is D
Solution and Explanation
The given matrices are: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}. \]
1. Equating Corresponding Elements: - From the first row, first column: \[ x + y = 6. \] - From the second row, second column: \[ xy = 8. \]
2. Expression to Evaluate: We need to find: \[ \frac{24}{x} + \frac{24}{y}. \]
Simplify the expression using the identity \( \frac{a}{x} + \frac{a}{y} = a \cdot \frac{x + y}{xy} \): \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{x + y}{xy}. \]
3. Substitute Known Values: - \( x + y = 6 \), - \( xy = 8 \). Substitute these values: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{6}{8}. \]
Simplify: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{3}{4} = 18. \]
Hence, the value of \( \frac{24}{x} + \frac{24}{y} \) is (D) 18.
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