If `x^a y^b=e^m , x^c y^d=e^n ,Delta_1=|(m,b),(n,d)|,and Delta_2 =|(a,m),(c,n)| and Delta_3=|(a,b),(c,d)|,` then the values of x and y are

Options

A. `(Delta_(1))/(Delta_(3)) and (Delta_(2))/(Delta_(3))`
B. `(Delta_(2))/(Delta_(1)) and (Delta_(3))/(Delta_(1))`
C. `log ((Delta_(1))/(Delta_(3))), log ((Delta_(2))/(Delta_(3)))`
D. `e^(Delta_(1)//Delta_(3)) and e^(Delta_(2)//Delta_(3))`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Write the given equations
We start with the two equations provided in the problem:
1. \( x^a y^b = e^m \)
2. \( x^c y^d = e^n \)

### Step 2: Take the logarithm of both sides
Taking the natural logarithm of both equations, we apply the logarithmic property that states \( \log(a^b) = b \log(a) \):
1. From \( x^a y^b = e^m \):
\[ a \log x + b \log y = m \]
2. From \( x^c y^d = e^n \):
\[ c \log x + d \log y = n \]

### Step 3: Set up the system of equations
We now have a system of linear equations:
1. \( a \log x + b \log y = m \) (Equation 1)
2. \( c \log x + d \log y = n \) (Equation 2)

### Step 4: Use Cramer’s Rule
To solve for \( \log x \) and \( \log y \) using Cramer's rule, we can express the equations in matrix form:
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \log x \\ \log y \end{pmatrix} = \begin{pmatrix} m \\ n \end{pmatrix} \]

### Step 5: Calculate determinants
We need to calculate the determinants for Cramer’s rule:
1. The determinant of the coefficient matrix:
\[ \Delta_3 = |(a, b), (c, d)| = ad - bc \]
2. The determinant for \( \log x \):
\[ \Delta_1 = |(m, b), (n, d)| = md - bn \]
3. The determinant for \( \log y \):
\[ \Delta_2 = |(a, m), (c, n)| = an - mc \]

### Step 6: Solve for \( \log x \) and \( \log y \)
Using Cramer’s rule, we have:
1. For \( \log x \):
\[ \log x = \frac{\Delta_1}{\Delta_3} = \frac{md - bn}{ad - bc} \]
2. For \( \log y \):
\[ \log y = \frac{\Delta_2}{\Delta_3} = \frac{an - mc}{ad - bc} \]

### Step 7: Exponentiate to find \( x \) and \( y \)
Taking the exponential of both sides to solve for \( x \) and \( y \):
1. For \( x \):
\[ x = e^{\log x} = e^{\frac{\Delta_1}{\Delta_3}} = e^{\frac{md - bn}{ad - bc}} \]
2. For \( y \):
\[ y = e^{\log y} = e^{\frac{\Delta_2}{\Delta_3}} = e^{\frac{an - mc}{ad - bc}} \]

### Final Result
Thus, the values of \( x \) and \( y \) are:
\[ x = e^{\frac{\Delta_1}{\Delta_3}}, \quad y = e^{\frac{\Delta_2}{\Delta_3}} \]