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].
