If x^a y^b=e^m, x^c y^d=e^n, Delta_1=left|beg | vedclass

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(A) Taking the natural logarithm on both sides of the given equations:$a \ln x + b \ln y = m$$c \ln x + d \ln y = n$Let $X = \ln x$ and $Y = \ln y$. T

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$e^{\frac{\Delta_1}{\Delta_3}}$ and $e^{\frac{\Delta_2}{\Delta_3}}$

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(A) Taking the natural logarithm on both sides of the given equations:$a \ln x + b \ln y = m$$c \ln x + d \ln y = n$Let $X = \ln x$ and $Y = \ln y$. The system becomes:$aX + bY = m$$cX + dY = n$Using Cramer's Rule:$X = \frac{\Delta_1}{\Delta_3} = \frac{\begin{vmatrix} m & b \ n & d \end{vmatrix}}{\begin{vmatrix} a & b \ c & d \end{vmatrix}}$$Y = \frac{\Delta_2}{\Delta_3} = \frac{\begin{vmatrix} a & m \ c & n \end{vmatrix}}{\begin{vmatrix} a & b \ c & d \end{vmatrix}}$Since $X = \ln x$,we have $x = e^X = e^{\frac{\Delta_1}{\Delta_3}}$.Since $Y = \ln y$,we have $y = e^Y = e^{\frac{\Delta_2}{\Delta_3}}$.Thus,the values are $e^{\frac{\Delta_1}{\Delta_3}}$ and $e^{\frac{\Delta_2}{\Delta_3}}$.

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