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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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 $e^{\frac{\Delta_2}{\Delta_3}}$.

Similar Questions

If $\frac{5}{m}+\frac{2}{n}=9$ and $\frac{3}{m}+\frac{4}{n}=11$ and $mn \neq 0$,then the values of $m$ and $n$ are . . . . . . respectively.

For the system $x-y+z=4, 2x+y-3z=0, x+y+z=2$,the values of $x, y, z$ respectively are given by

The existence of a unique solution for the system of equations $x+y+z=\beta$,$5x-y+\alpha z=10$,and $2x+3y-z=6$ depends on:

The system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = \mu$ has no solution for:

Vedclass Test Series

Exam Paper Generator

Online Exam Module