To determine which system of linear equations can be solved using the given determinants, we analyze the matrices provided.
We are given:
[tex]\[
\left|A_x\right|=\left|\begin{array}{cc}
20 & -3 \\
-192 & 8
\end{array}\right|, \quad \left|A_y\right|=\left|\begin{array}{cc}
2 & 20 \\
12 & -192
\end{array}\right|
\][/tex]
where [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex] are determinants. We aim to identify the corresponding system of linear equations.
First, let's denote the systems of linear equations in question:
1. System 1:
[tex]\[
\begin{cases}
2x - 3y = 20 \\
12x + 8y = -192
\end{cases}
\][/tex]
2. System 2:
[tex]\[
\begin{cases}
20x - 3y = 2 \\
-192x + 8y = 12
\end{cases}
\][/tex]
3. System 3:
[tex]\[
\begin{cases}
-3x + 2y = 20 \\
8x + 12y = -192
\end{cases}
\][/tex]
4. System 4:
[tex]\[
\begin{cases}
-3x + 8y = 20
\end{cases}
\][/tex]
We calculate the determinants [tex]\( \left|A_x\right| \)[/tex] and [tex]\( \left|A_y\right| \)[/tex] and compare with given results.
### Determinant [tex]\( \left|A_x\right| \)[/tex]:
[tex]\[
\left|A_x\right| = \begin{vmatrix} 20 & -3 \\ -192 & 8 \end{vmatrix} = (20 \times 8) - (-192 \times -3) = 160 - 576 = -416
\][/tex]
### Determinant [tex]\( \left|A_y\right| \)[/tex]:
[tex]\[
\left|A_y\right| = \begin{vmatrix} 2 & 20 \\ 12 & -192 \end{vmatrix} = (2 \times -192) - (20 \times 12) = -384 - 240 = -624
\][/tex]
The calculated values of the determinants confirm the results:
[tex]\[ \left|A_x\right| = -416 \quad \left|A_y\right| = -624 \][/tex]
Comparing these determinant values with the systems:
- For System 2:
- The first equation [tex]\( 20x - 3y = 2 \)[/tex] forms matrix [tex]\( A_x \)[/tex] with coefficients [tex]\( 20 \)[/tex] and [tex]\(-3\)[/tex].
- The second equation [tex]\( -192x + 8y = 12 \)[/tex] aligns with [tex]\( A_x \)[/tex] as [tex]\( -192 \)[/tex] and [tex]\( 8 \)[/tex].
Thus, the system of linear equations related to the given determinants is:
[tex]\[
\begin{cases}
20x - 3y = 2 \\
-192x + 8y = 12
\end{cases}
\][/tex]
This system aligns with the given determinants and their corresponding calculations, confirming that System 2 is the correct set of equations.
