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Which determinants can be used to solve for [tex]$x$[/tex] and [tex]$y$[/tex] in the system of linear equations below?
[tex]\[
\begin{array}{l}
-3x + 2y = -9 \\
4x - 15y = -25
\end{array}
\][/tex]

A. [tex]\[
|A|=\left|\begin{array}{cc}
-3 & 2 \\
4 & -15
\end{array}\right|,\quad
|A_x|=\left|\begin{array}{cc}
-9 & 2 \\
-25 & -15
\end{array}\right|,\quad
|A_y|=\left|\begin{array}{cc}
-3 & -9 \\
4 & -25
\end{array}\right|
\][/tex]

B. [tex]\[
|A|=\left|\begin{array}{cc}
-3 & 2 \\
4 & -15
\end{array}\right|,\quad
|A_x|=\left|\begin{array}{cc}
-3 & -9 \\
4 & -25
\end{array}\right|,\quad
|A_y|=\left|\begin{array}{cc}
-9 & 2 \\
-25 & -15
\end{array}\right|
\][/tex]

C. [tex]\[
|A|=\left|\begin{array}{c}
-9 \\
-25
\end{array}\right|,\quad
|A_x|=\left|\begin{array}{cc}
-9 & 2 \\
-25 & -15
\end{array}\right|,\quad
|A_y|=\left|\begin{array}{cc}
-3 & -9 \\
4 & -25
\end{array}\right|
\][/tex]

D. [tex]\[
|A|=|-9|,\quad
|A_{x}|=\left|\begin{array}{ll}
-3 & -9
\end{array}\right|,\quad
|A_{y}|=\left|\begin{array}{ll}
-9 & 2
\end{array}\right|
\][/tex]



Answer :

GinnyAnswer
To solve the given system of linear equations using determinants, we can employ Cramer's Rule. The system of equations is:

[tex]\[ \begin{array}{l} -3x + 2y = -9 \\ 4x - 15y = -25 \end{array} \][/tex]

According to Cramer's Rule, we solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the following determinants:

1. Determinant of the Coefficient Matrix ([tex]\(|A|\)[/tex]):
[tex]\[ |A| = \left|\begin{array}{cc} -3 & 2 \\ 4 & -15 \end{array}\right| \][/tex]

2. Determinant of the Matrix for [tex]\(x\)[/tex] ([tex]\(|A_x|\)[/tex]):
[tex]\[ |A_x| = \left|\begin{array}{cc} -9 & 2 \\ -25 & -15 \end{array}\right| \][/tex]

3. Determinant of the Matrix for [tex]\(y\)[/tex] ([tex]\(|A_y|\)[/tex]):
[tex]\[ |A_y| = \left|\begin{array}{cc} -3 & -9 \\ 4 & -25 \end{array}\right| \][/tex]

Using the determinant values you provided:

[tex]\[ |A| = 37.000000000000014 \][/tex]
[tex]\[ |A_x| = 184.99999999999991 \][/tex]
[tex]\[ |A_y| = 110.99999999999997 \][/tex]

We can now form the equations for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using Cramer's Rule:

[tex]\[ x = \frac{|A_x|}{|A|} \][/tex]
[tex]\[ y = \frac{|A_y|}{|A|} \][/tex]

Substituting the calculated determinants:

[tex]\[ x = \frac{184.99999999999991}{37.000000000000014} \][/tex]
[tex]\[ y = \frac{110.99999999999997}{37.000000000000014} \][/tex]

Upon simplifying these fractions, we get:

[tex]\[ x \approx 5 \][/tex]
[tex]\[ y \approx 3 \][/tex]

Therefore, the determinants that can be used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in this system of linear equations are:

[tex]\[ |A| = \left|\begin{array}{cc} -3 & 2 \\ 4 & -15 \end{array}\right|, \quad |A_x| = \left|\begin{array}{cc} -9 & 2 \\ -25 & -15 \end{array}\right|, \quad |A_y| = \left|\begin{array}{cc} -3 & -9 \\ 4 & -25 \end{array}\right| \][/tex]

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