Let's solve for the determinants of the given linear system systematically.
We are given the linear system:
[tex]\[
\begin{array}{l}
5x + 2y = 14 \\
-3x - 5y = 3
\end{array}
\][/tex]
Step 1: Write the system in matrix form:
This system can be represented in the matrix form [tex]\(A \mathbf{x} = \mathbf{b}\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(\mathbf{x}\)[/tex] is the vector of variables, and [tex]\(\mathbf{b}\)[/tex] is the constants vector.
[tex]\[
A = \begin{pmatrix}
5 & 2 \\
-3 & -5
\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix}
14 \\
3
\end{pmatrix}
\][/tex]
Step 2: Find the determinant of matrix [tex]\(A\)[/tex]:
[tex]\[
|A| = \begin{vmatrix}
5 & 2 \\
-3 & -5
\end{vmatrix}
\][/tex]
Using the formula for the determinant of a 2x2 matrix, [tex]\(|A| = ad - bc\)[/tex]:
[tex]\[
|A| = (5 \cdot -5) - (2 \cdot -3) = -25 + 6 = -19
\][/tex]
So, the determinant of [tex]\(A\)[/tex] is:
[tex]\[
|A| = -19
\][/tex]
Step 3: Find the determinant of matrix [tex]\(A_x\)[/tex]:
Matrix [tex]\(A_x\)[/tex] is obtained by replacing the first column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[
A_x = \begin{pmatrix}
14 & 2 \\
3 & -5
\end{pmatrix}
\][/tex]
Now, we compute the determinant of [tex]\(A_x\)[/tex]:
[tex]\[
|A_x| = \begin{vmatrix}
14 & 2 \\
3 & -5
\end{vmatrix}
\][/tex]
[tex]\[
|A_x| = (14 \cdot -5) - (2 \cdot 3) = -70 - 6 = -76
\][/tex]
So, the determinant of [tex]\(A_x\)[/tex] is:
[tex]\[
|A_x| = -76
\][/tex]
Step 4: Find the determinant of matrix [tex]\(A_y\)[/tex]:
Matrix [tex]\(A_y\)[/tex] is obtained by replacing the second column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[
A_y = \begin{pmatrix}
5 & 14 \\
-3 & 3
\end{pmatrix}
\][/tex]
Now, we compute the determinant of [tex]\(A_y\)[/tex]:
[tex]\[
|A_y| = \begin{vmatrix}
5 & 14 \\
-3 & 3
\end{vmatrix}
\][/tex]
[tex]\[
|A_y| = (5 \cdot 3) - (14 \cdot -3) = 15 + 42 = 57
\][/tex]
So, the determinant of [tex]\(A_y\)[/tex] is:
[tex]\[
|A_y| = 57
\][/tex]
In summary, the determinants for the given linear system are:
[tex]\[
\begin{array}{l}
|A| = -19 \\
|A_x| = -76 \\
|A_y| = 57
\end{array}
\][/tex]
