Solve the following system of equation by Cramer's rule.
x+y=4 and 3x-2y=9

Answer

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

Given the equations:
1. \( x + y = 4 \) (Equation 1)
2. \( 3x - 2y = 9 \) (Equation 2)

### Step 1: Write the equations in standard form
The equations are already in standard form.

### Step 2: Identify the coefficients and constants
From the equations, we can identify the coefficients:
- Coefficients of \( x \) and \( y \) from Equation 1: \( a_1 = 1, b_1 = 1 \)
- Coefficients of \( x \) and \( y \) from Equation 2: \( a_2 = 3, b_2 = -2 \)
- Constants: \( c_1 = 4, c_2 = 9 \)

### Step 3: Form the coefficient matrix and calculate the determinant \( D \)
The coefficient matrix is:
\[ D = \begin{vmatrix} 1 & 1 \\ 3 & -2 \end{vmatrix} \]

Calculating the determinant \( D \):
\[ D = (1)(-2) - (1)(3) = -2 - 3 = -5 \]

### Step 4: Calculate \( D_1 \) and \( D_2 \)
**For \( D_1 \)** (replace the first column with constants):
\[ D_1 = \begin{vmatrix} 4 & 1 \\ 9 & -2 \end{vmatrix} \]
Calculating \( D_1 \):
\[ D_1 = (4)(-2) - (1)(9) = -8 - 9 = -17 \]

**For \( D_2 \)** (replace the second column with constants):
\[ D_2 = \begin{vmatrix} 1 & 4 \\ 3 & 9 \end{vmatrix} \]
Calculating \( D_2 \):
\[ D_2 = (1)(9) - (4)(3) = 9 - 12 = -3 \]

### Step 5: Calculate \( x \) and \( y \)
Using Cramer's rule:
\[ x = \frac{D_1}{D} = \frac{-17}{-5} = \frac{17}{5} \]

\[ y = \frac{D_2}{D} = \frac{-3}{-5} = \frac{3}{5} \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = \frac{17}{5}, \quad y = \frac{3}{5} \]