Solve the simultaneous equations using Cramer's rule : `3x-4y = 7, 5x +2y=3`

Answer

To solve the simultaneous equations using Cramer's rule, we will follow these steps:

Given equations:
1. \(3x - 4y = 7\) (Equation 1)
2. \(5x + 2y = 3\) (Equation 2)

### Step 1: Identify coefficients and constants
From the equations, we can identify:
- \(A_1 = 3\), \(B_1 = -4\), \(C_1 = 7\) (from Equation 1)
- \(A_2 = 5\), \(B_2 = 2\), \(C_2 = 3\) (from Equation 2)

### Step 2: Calculate the determinant \(D\)
The determinant \(D\) is given by:
\[ D = \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix} = \begin{vmatrix} 3 & -4 \\ 5 & 2 \end{vmatrix} \]

Calculating \(D\):
\[ D = (3 \cdot 2) - (-4 \cdot 5) = 6 + 20 = 26 \]

### Step 3: Calculate the determinant \(D_x\)
The determinant \(D_x\) is given by:
\[ D_x = \begin{vmatrix} C_1 & B_1 \\ C_2 & B_2 \end{vmatrix} = \begin{vmatrix} 7 & -4 \\ 3 & 2 \end{vmatrix} \]

Calculating \(D_x\):
\[ D_x = (7 \cdot 2) - (-4 \cdot 3) = 14 + 12 = 26 \]

### Step 4: Calculate the determinant \(D_y\)
The determinant \(D_y\) is given by:
\[ D_y = \begin{vmatrix} A_1 & C_1 \\ A_2 & C_2 \end{vmatrix} = \begin{vmatrix} 3 & 7 \\ 5 & 3 \end{vmatrix} \]

Calculating \(D_y\):
\[ D_y = (3 \cdot 3) - (7 \cdot 5) = 9 - 35 = -26 \]

### Step 5: Calculate \(x\) and \(y\)
Using Cramer's rule:
\[ x = \frac{D_x}{D} = \frac{26}{26} = 1 \]
\[ y = \frac{D_y}{D} = \frac{-26}{26} = -1 \]

### Final Solution
The solution to the simultaneous equations is:
\[ x = 1, \quad y = -1 \]

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