To identify the determinants for the given linear system:
[tex]\[
\begin{array}{l}
5x + 2y = 14 \\
-3x - 5y = 3
\end{array}
\][/tex]
we need to perform the following steps:
1. Identify the Coefficients and Constants:
From the given system, the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] and the constants are:
[tex]\[
\begin{array}{l}
a_{11} = 5, \quad a_{12} = 2, \quad b_1 = 14 \\
a_{21} = -3, \quad a_{22} = -5, \quad b_2 = 3
\end{array}
\][/tex]
2. Calculate the Determinant of Matrix [tex]\(A\)[/tex]:
Matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \begin{bmatrix}
5 & 2 \\
-3 & -5
\end{bmatrix}
\][/tex]
The determinant of [tex]\(A\)[/tex], denoted as [tex]\(|A|\)[/tex], is calculated using the formula:
[tex]\[
|A| = a_{11}a_{22} - a_{12}a_{21}
\][/tex]
Substituting the values:
[tex]\[
|A| = (5)(-5) - (2)(-3) = -25 + 6 = -19
\][/tex]
Therefore,
[tex]\[
|A| = -19
\][/tex]
3. Calculate 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 constants [tex]\(b_1\)[/tex] and [tex]\(b_2\)[/tex]:
[tex]\[
A_x = \begin{bmatrix}
14 & 2 \\
3 & -5
\end{bmatrix}
\][/tex]
The determinant of [tex]\(A_x\)[/tex], denoted as [tex]\(|A_x|\)[/tex], is calculated using the formula:
[tex]\[
|A_x| = b_1a_{22} - a_{12}b_2
\][/tex]
Substituting the values:
[tex]\[
|A_x| = (14)(-5) - (2)(3) = -70 - 6 = -76
\][/tex]
Therefore,
[tex]\[
|A_x| = -76
\][/tex]
4. Calculate 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 constants [tex]\(b_1\)[/tex] and [tex]\(b_2\)[/tex]:
[tex]\[
A_y = \begin{bmatrix}
5 & 14 \\
-3 & 3
\end{bmatrix}
\][/tex]
The determinant of [tex]\(A_y\)[/tex], denoted as [tex]\(|A_y|\)[/tex], is calculated using the formula:
[tex]\[
|A_y| = a_{11}b_2 - b_1a_{21}
\][/tex]
Substituting the values:
[tex]\[
|A_y| = (5)(3) - (14)(-3) = 15 + 42 = 57
\][/tex]
Therefore,
[tex]\[
|A_y| = 57
\][/tex]
So, the determinants for the given linear system are:
[tex]\[
|A| = -19
\][/tex]
[tex]\[
\left|A_x\right| = -76
\][/tex]
[tex]\[
\left|A_y\right| = 57
\][/tex]
