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

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

### Step 2: Identify coefficients and constants
From the equations, we identify the coefficients of \(x\) and \(y\):
- Coefficients:
- For \(x\): \(1\) (from Equation 1) and \(3\) (from Equation 2)
- For \(y\): \(1\) (from Equation 1) and \(-2\) (from Equation 2)
- Constants: \(4\) (from Equation 1) and \(9\) (from Equation 2)

### Step 3: Form the coefficient matrix and calculate its determinant (D)
The coefficient matrix \(A\) is:
\[ A = \begin{pmatrix} 1 & 1 \\ 3 & -2 \end{pmatrix} \]
Now, we calculate the determinant \(D\):
\[ D = \begin{vmatrix} 1 & 1 \\ 3 & -2 \end{vmatrix} = (1)(-2) - (1)(3) = -2 - 3 = -5 \]

### Step 4: Form the matrices for \(D_1\) and \(D_2\)

#### Calculate \(D_1\)
Replace the first column of the coefficient matrix with the constants:
\[ D_1 = \begin{vmatrix} 4 & 1 \\ 9 & -2 \end{vmatrix} = (4)(-2) - (1)(9) = -8 - 9 = -17 \]

#### Calculate \(D_2\)
Replace the second column of the coefficient matrix with the constants:
\[ D_2 = \begin{vmatrix} 1 & 4 \\ 3 & 9 \end{vmatrix} = (1)(9) - (4)(3) = 9 - 12 = -3 \]

### Step 5: Calculate \(x\) and \(y\) using Cramer's Rule
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
The solution to the system of equations is:
\[ x = \frac{17}{5}, \quad y = \frac{3}{5} \]

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