To identify the determinants for the given linear system:
1. System of Equations:
[tex]\[ 5x + 2y = 14 \][/tex]
[tex]\[ -3x - 5y = 3 \][/tex]
2. Matrix Representation:
The coefficient matrix [tex]\( A \)[/tex] and the constants vector [tex]\( B \)[/tex] are:
[tex]\[ A = \begin{pmatrix} 5 & 2 \\ -3 & -5 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} 14 \\ 3 \end{pmatrix} \][/tex]
3. Determinant of [tex]\(A\)[/tex]:
For a 2x2 matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex], the determinant is given by:
[tex]\[ \det(A) = ad - bc \][/tex]
Plugging in the values from matrix [tex]\(A\)[/tex]:
[tex]\[ \det(A) = (5 \times -5) - (2 \times -3) \][/tex]
[tex]\[ \det(A) = -25 - (-6) \][/tex]
[tex]\[ \det(A) = -25 + 6 \][/tex]
[tex]\[ \det(A) = -19 \][/tex]
4. Matrix [tex]\( A_x \)[/tex]:
To find [tex]\( \det(A_x) \)[/tex], replace the first column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[ A_x = \begin{pmatrix} 14 & 2 \\ 3 & -5 \end{pmatrix} \][/tex]
The determinant is:
[tex]\[ \det(A_x) = (14 \times -5) - (2 \times 3) \][/tex]
[tex]\[ \det(A_x) = -70 - 6 \][/tex]
[tex]\[ \det(A_x) = -76 \][/tex]
5. Matrix [tex]\( A_y \)[/tex]:
To find [tex]\( \det(A_y) \)[/tex], replace the second column of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
[tex]\[ A_y = \begin{pmatrix} 5 & 14 \\ -3 & 3 \end{pmatrix} \][/tex]
The determinant is:
[tex]\[ \det(A_y) = (5 \times 3) - (14 \times -3) \][/tex]
[tex]\[ \det(A_y) = 15 - (-42) \][/tex]
[tex]\[ \det(A_y) = 15 + 42 \][/tex]
[tex]\[ \det(A_y) = 57 \][/tex]
Therefore, the determinants for the given linear system are:
[tex]\[
|A| = -19
\][/tex]
[tex]\[
|A_x| = -76
\][/tex]
[tex]\[
|A_y| = 57
\][/tex]
