Using Cramer's Rule , solve the following system of equations :
`3x-2y=5`, `x-3y=-3`

Answer

To solve the system of equations using Cramer's Rule, we will follow these steps:

Given the equations:
1. \( 3x - 2y = 5 \) (Equation 1)
2. \( x - 3y = -3 \) (Equation 2)

### Step 1: Write the system in matrix form
The system of equations can be represented in the form \( AX = B \), where:
\[ A = \begin{bmatrix} 3 & -2 \\ 1 & -3 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ -3 \end{bmatrix} \]

### Step 2: Calculate the determinant \( D \)
The determinant \( D \) of matrix \( A \) is calculated as follows:
\[ D = \begin{vmatrix} 3 & -2 \\ 1 & -3 \end{vmatrix} = (3)(-3) - (-2)(1) = -9 + 2 = -7 \]

### Step 3: Calculate \( D_x \)
To find \( D_x \), we replace the first column of \( A \) with the column matrix \( B \):
\[ D_x = \begin{vmatrix} 5 & -2 \\ -3 & -3 \end{vmatrix} = (5)(-3) - (-2)(-3) = -15 - 6 = -21 \]

### Step 4: Calculate \( D_y \)
To find \( D_y \), we replace the second column of \( A \) with the column matrix \( B \):
\[ D_y = \begin{vmatrix} 3 & 5 \\ 1 & -3 \end{vmatrix} = (3)(-3) - (5)(1) = -9 - 5 = -14 \]

### Step 5: Calculate \( x \) and \( y \)
Using Cramer's Rule:
\[ x = \frac{D_x}{D} = \frac{-21}{-7} = 3 \]
\[ y = \frac{D_y}{D} = \frac{-14}{-7} = 2 \]

### Final Solution
The solution to the system of equations is:
\[ x = 3, \quad y = 2 \]