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]
