3. Use Cramer's rule to solve the following system of equations:

[tex]\[
\begin{array}{l}
5x + 3y = 7 \\
4x + 5y = 3
\end{array}
\][/tex]

A. [tex]\( x = -2, y = -1 \)[/tex]

B. [tex]\( x = 4, y = 1 \)[/tex]

C. [tex]\( x = 2, y = 5 \)[/tex]

D. [tex]\( x = 2, y = -1 \)[/tex]



Answer :

To solve the given system of equations using Cramer's rule, we will follow a systematic approach. Cramer's rule involves the use of determinants of matrices to find the solution to a system of linear equations. Here's the step-by-step procedure:

[tex]\[ \begin{array}{l} 5x + 3y = 7 \\ 4x + 5y = 3 \end{array} \][/tex]

### Step 1: Construct the Coefficient Matrix and the Constant Matrix

The coefficient matrix [tex]\( A \)[/tex] and the constant matrix [tex]\( B \)[/tex] are defined as follows:

[tex]\[ A = \begin{bmatrix} 5 & 3 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 7 \\ 3 \end{bmatrix} \][/tex]

### Step 2: Calculate the Determinant of the Coefficient Matrix

The determinant ([tex]\(\text{det}(A)\)[/tex]) of matrix [tex]\( A \)[/tex] is calculated as:

[tex]\[ \text{det}(A) = \begin{vmatrix} 5 & 3 \\ 4 & 5 \end{vmatrix} = (5 \cdot 5) - (3 \cdot 4) = 25 - 12 = 13 \][/tex]

### Step 3: Construct Matrices for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]

To find [tex]\( x \)[/tex]:

We create matrix [tex]\( A_x \)[/tex] by substituting the constants from matrix [tex]\( B \)[/tex] into the [tex]\( x \)[/tex]-column of matrix [tex]\( A \)[/tex]:

[tex]\[ A_x = \begin{bmatrix} 7 & 3 \\ 3 & 5 \end{bmatrix} \][/tex]

To find [tex]\( y \)[/tex]:

We create matrix [tex]\( A_y \)[/tex] by substituting the constants from matrix [tex]\( B \)[/tex] into the [tex]\( y \)[/tex]-column of matrix [tex]\( A \)[/tex]:

[tex]\[ A_y = \begin{bmatrix} 5 & 7 \\ 4 & 3 \end{bmatrix} \][/tex]

### Step 4: Calculate the Determinants of [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex]

1. The determinant ([tex]\(\text{det}(A_x)\)[/tex]) of matrix [tex]\( A_x \)[/tex] is:

[tex]\[ \text{det}(A_x) = \begin{vmatrix} 7 & 3 \\ 3 & 5 \end{vmatrix} = (7 \cdot 5) - (3 \cdot 3) = 35 - 9 = 26 \][/tex]

2. The determinant ([tex]\(\text{det}(A_y)\)[/tex]) of matrix [tex]\( A_y \)[/tex] is:

[tex]\[ \text{det}(A_y) = \begin{vmatrix} 5 & 7 \\ 4 & 3 \end{vmatrix} = (5 \cdot 3) - (7 \cdot 4) = 15 - 28 = -13 \][/tex]

### Step 5: Calculate [tex]\( x \)[/tex] and [tex]\( y \)[/tex] Using Cramer's Rule

1. [tex]\( x \)[/tex] is given by:

[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{26}{13} = 2 \][/tex]

2. [tex]\( y \)[/tex] is given by:

[tex]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-13}{13} = -1 \][/tex]

### Step 6: Write the Solution

The solution to the system of equations is:

[tex]\[ x = 2, \quad y = -1 \][/tex]

Thus, the correct choice is:
[tex]\[ D. \quad x = 2, y = -1 \][/tex]