Sure, let's solve the given system of linear equations using Cramer's Rule.
The system of equations is:
[tex]\[
\begin{cases}
y + 2x - 19 = 0 \\
2x - 3y + 3 = 0
\end{cases}
\][/tex]
First, rearrange the equations to the standard form [tex]\(ax + by + c = 0\)[/tex]:
1. [tex]\(2x - 3y + 3 = 0\)[/tex]
2. [tex]\(2x + y - 19 = 0\)[/tex]
We can extract the coefficients from these equations:
[tex]\[
\begin{cases}
a_1 = 2, \quad b_1 = -3, \quad c_1 = 3 \\
a_2 = 2, \quad b_2 = 1, \quad c_2 = -19
\end{cases}
\][/tex]
Cramer's Rule uses determinants to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. First, we need to find the determinant [tex]\(D\)[/tex] of the coefficient matrix:
[tex]\[
D = \begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix} = \begin{vmatrix}
2 & -3 \\
2 & 1
\end{vmatrix}
\][/tex]
Calculating [tex]\(D\)[/tex]:
[tex]\[
D = (2 \cdot 1) - (2 \cdot -3) = 2 + 6 = 8
\][/tex]
Next, we find the determinant [tex]\(D_x\)[/tex] by replacing the first column of the coefficient matrix with the constants from the right side of the equations:
[tex]\[
D_x = \begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2
\end{vmatrix} = \begin{vmatrix}
3 & -3 \\
-19 & 1
\end{vmatrix}
\][/tex]
Calculating [tex]\(D_x\)[/tex]:
[tex]\[
D_x = (3 \cdot 1) - (-19 \cdot -3) = 3 - 57 = -54
\][/tex]
Then, we find the determinant [tex]\(D_y\)[/tex] by replacing the second column of the coefficient matrix with the constants from the right side of the equations:
[tex]\[
D_y = \begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix} = \begin{vmatrix}
2 & 3 \\
2 & -19
\end{vmatrix}
\][/tex]
Calculating [tex]\(D_y\)[/tex]:
[tex]\[
D_y = (2 \cdot -19) - (2 \cdot 3) = -38 - 6 = -44
\][/tex]
Using Cramer's Rule, we can now solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
x = \frac{D_x}{D} = \frac{-54}{8} = -6.75
\][/tex]
[tex]\[
y = \frac{D_y}{D} = \frac{-44}{8} = -5.5
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -6.75, \quad y = -5.5
\][/tex]
These are the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
