To solve for [tex]\( y \)[/tex] using Cramer's Rule, follow these steps:
1. Write the system of equations:
[tex]\[
\begin{cases}
9x - 2y = 5 \\
-3x - 4y = -4
\end{cases}
\][/tex]
2. Determine the coefficient matrix [tex]\( A \)[/tex] and the constants matrix:
[tex]\[
A = \begin{pmatrix}
9 & -2 \\
-3 & -4
\end{pmatrix}
,\quad B = \begin{pmatrix}
5 \\
-4
\end{pmatrix}
\][/tex]
3. Calculate the determinant [tex]\( D \)[/tex] of matrix [tex]\( A \)[/tex]:
[tex]\[
D = \begin{vmatrix}
9 & -2 \\
-3 & -4
\end{vmatrix}
= (9)(-4) - (-2)(-3)
= -36 - 6
= -42
\][/tex]
4. Calculate the determinant [tex]\( D_y \)[/tex] for the [tex]\( y \)[/tex]-component by replacing the [tex]\( y \)[/tex]-column in [tex]\( A \)[/tex] with the constants matrix [tex]\( B \)[/tex]:
[tex]\[
D_y = \begin{vmatrix}
9 & 5 \\
-3 & -4
\end{vmatrix}
= (9)(-4) - (5)(-3)
= -36 + 15
= -21
\][/tex]
5. Calculate [tex]\( y \)[/tex] using Cramer's Rule:
[tex]\[
y = \frac{D_y}{D} = \frac{-21}{-42} = \frac{1}{2}
\][/tex]
Therefore, the value of [tex]\( y \)[/tex] in the solution to the system of linear equations is [tex]\( \frac{1}{2} \)[/tex].
