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Using Cramer's Rule, what is the value of [tex]$y$[/tex] in the solution to the system of linear equations below?

[tex]\[
\begin{array}{l}
9x - 2y = 5 \\
-3x - 4y = -4
\end{array}
\][/tex]

A. [tex]\frac{\left|\begin{array}{cc}9 & 5 \\ -3 & -4\end{array}\right|}{-42} = \frac{2}{3}[/tex]

B. [tex]\frac{\left|\begin{array}{cc}9 & 5 \\ -3 & -4\end{array}\right|}{-42} = \frac{1}{2}[/tex]

C. [tex]\frac{\left|\begin{array}{cc}5 & -2 \\ -4 & -4\end{array}\right|}{-42} = \frac{2}{3}[/tex]



Answer :

GinnyAnswer
To find the value of [tex]\( y \)[/tex] using Cramer's Rule for the given system of linear equations:

[tex]\[ \begin{array}{rcl} 9x - 2y &=& 5 \\ -3x - 4y &=& -4 \end{array} \][/tex]

we proceed with the following steps:

1. Write the coefficient matrix:

[tex]\[ \begin{pmatrix} 9 & -2 \\ -3 & -4 \end{pmatrix} \][/tex]

2. Calculate the determinant of the coefficient matrix:

[tex]\[ \text{det} = \begin{vmatrix} 9 & -2 \\ -3 & -4 \end{vmatrix} = (9 \cdot -4) - (-2 \cdot -3) = -36 - 6 = -42 \][/tex]

3. Construct the matrices required to find [tex]\( y \)[/tex] by replacing the column of [tex]\( y \)[/tex]-coefficients with the constants:

[tex]\[ \text{Matrix for finding } y = \begin{pmatrix} 9 & 5 \\ -3 & -4 \end{pmatrix} \][/tex]

4. Calculate the determinant of this new matrix:

[tex]\[ \text{det}_y = \begin{vmatrix} 9 & 5 \\ -3 & -4 \end{vmatrix} = (9 \cdot -4) - (5 \cdot -3) = -36 + 15 = -21 \][/tex]

5. Find the value of [tex]\( y \)[/tex] using Cramer's Rule which states that [tex]\( y = \frac{\text{det}_y}{\text{det}} \)[/tex]:

[tex]\[ y = \frac{-21}{-42} = \frac{1}{2} \][/tex]

So, the value of [tex]\( y \)[/tex] in the solution to the system of linear equations is [tex]\( \boxed{0.5} \)[/tex].

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