With the help of determinants , solve the following systems of equations :
`2x+3y=9`, `3x-2y=7`

Answer

To solve the system of equations using determinants, we will follow the steps outlined below:

### Given Equations:
1. \( 2x + 3y = 9 \) (Equation 1)
2. \( 3x - 2y = 7 \) (Equation 2)

### Step 1: Write the equations in the form of \( Ax = B \)
We can express the system of equations in matrix form \( Ax = B \), where:
- \( A \) is the coefficient matrix,
- \( x \) is the variable matrix,
- \( B \) is the constant matrix.

Here, we have:
\[ A = \begin{pmatrix} 2 & 3 \\ 3 & -2 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 9 \\ 7 \end{pmatrix} \]

### Step 2: Calculate the determinant \( \Delta \) of matrix \( A \)
The determinant \( \Delta \) is calculated as follows:
\[ \Delta = \begin{vmatrix} 2 & 3 \\ 3 & -2 \end{vmatrix} = (2 \cdot -2) - (3 \cdot 3) = -4 - 9 = -13 \]

### Step 3: Calculate \( \Delta_x \) (determinant when replacing the first column with \( B \))
To find \( \Delta_x \), we replace the first column of \( A \) with the matrix \( B \):
\[ \Delta_x = \begin{vmatrix} 9 & 3 \\ 7 & -2 \end{vmatrix} = (9 \cdot -2) - (3 \cdot 7) = -18 - 21 = -39 \]

### Step 4: Calculate \( \Delta_y \) (determinant when replacing the second column with \( B \))
To find \( \Delta_y \), we replace the second column of \( A \) with the matrix \( B \):
\[ \Delta_y = \begin{vmatrix} 2 & 9 \\ 3 & 7 \end{vmatrix} = (2 \cdot 7) - (9 \cdot 3) = 14 - 27 = -13 \]

### Step 5: Calculate \( x \) and \( y \) using Cramer’s Rule
Using Cramer’s Rule, we can find \( x \) and \( y \) as follows:
\[ x = \frac{\Delta_x}{\Delta} = \frac{-39}{-13} = 3 \]
\[ y = \frac{\Delta_y}{\Delta} = \frac{-13}{-13} = 1 \]

### Final Solution:
The solution to the system of equations is:
\[ x = 3, \quad y = 1 \]