To determine the value of [tex]\( x \)[/tex] in the given system of linear equations using Cramer's Rule, we start with these equations:
[tex]\[
5x - 4 = 3y
\][/tex]
[tex]\[
2x + 32 = 4y
\][/tex]
First, we reformulate these equations to standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[ 5x - 3y = 4 \][/tex]
[tex]\[ 2x - 4y = -32 \][/tex]
Next, we identify the coefficients and constants:
[tex]\[
\begin{aligned}
&\text{Coefficients of } x: 5, 2 \\
&\text{Coefficients of } y: -3, -4 \\
&\text{Constants: 4, -32}
\end{aligned}
\][/tex]
Using Cramer's Rule, we need to calculate the determinants:
[tex]\[ D = \begin{vmatrix} 5 & -3 \\ 2 & -4 \end{vmatrix} \][/tex]
[tex]\[ D_x = \begin{vmatrix} 4 & -3 \\ -32 & -4 \end{vmatrix} \][/tex]
[tex]\[ D_y = \begin{vmatrix} 5 & 4 \\ 2 & -32 \end{vmatrix} \][/tex]
To find [tex]\( D \)[/tex]:
[tex]\[ D = 5(-4) - (-3)(2) \][/tex]
[tex]\[ D = -20 + 6 \][/tex]
[tex]\[ D = -14 \][/tex]
To find [tex]\( D_x \)[/tex]:
[tex]\[ D_x = 4(-4) - (-32)(-3) \][/tex]
[tex]\[ D_x = -16 - 96 \][/tex]
[tex]\[ D_x = -112 \][/tex]
To find [tex]\( D_y \)[/tex]:
[tex]\[ D_y = 5(-32) - 4(2) \][/tex]
[tex]\[ D_y = -160 - 8 \][/tex]
[tex]\[ D_y = -168 \][/tex]
Using Cramer's Rule:
[tex]\[ x = \frac{D_x}{D} = \frac{-112}{-14} = 8 \][/tex]
Thus, the value of [tex]\( x \)[/tex] in the solution to the system of equations is:
[tex]\[ \boxed{8} \][/tex]
