Certainly! Let's solve the system of equations using Cramer's Rule step-by-step. Our system of equations is:
[tex]\[
\left\{
\begin{array}{l}
-2x + 4y - 9z = -32 \\
-3x + 5y - 9z = -30 \\
-x + 2y - 3z = -10
\end{array}
\right.
\][/tex]
### Step 1: Write down the coefficient matrix [tex]\( A \)[/tex]
The coefficient matrix [tex]\( A \)[/tex] is the matrix of the coefficients of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[
A = \begin{pmatrix}
-2 & 4 & -9 \\
-3 & 5 & -9 \\
-1 & 2 & -3 \\
\end{pmatrix}
\][/tex]
### Step 2: Calculate the determinant [tex]\( D \)[/tex] of matrix [tex]\( A \)[/tex]
To find [tex]\( D \)[/tex]:
[tex]\[
D = \det(A)
\][/tex]
[tex]\[
D = \begin{vmatrix}
-2 & 4 & -9 \\
-3 & 5 & -9 \\
-1 & 2 & -3
\end{vmatrix}
\][/tex]
### Step 3: Write down the matrices [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
These matrices are formed by replacing one column of [tex]\( A \)[/tex] with the constants from the right-hand side of the equations (-32, -30, -10).
Matrix [tex]\( D_x \)[/tex]: replace the [tex]\( x \)[/tex]-column with the constants
[tex]\[
D_x = \begin{pmatrix}
-32 & 4 & -9 \\
-30 & 5 & -9 \\
-10 & 2 & -3 \\
\end{pmatrix}
\][/tex]
Matrix [tex]\( D_y \)[/tex]: replace the [tex]\( y \)[/tex]-column with the constants
[tex]\[
D_y = \begin{pmatrix}
-2 & -32 & -9 \\
-3 & -30 & -9 \\
-1 & -10 & -3 \\
\end{pmatrix}
\][/tex]
Matrix [tex]\( D_z \)[/tex]: replace the [tex]\( z \)[/tex]-column with the constants
[tex]\[
D_z = \begin{pmatrix}
-2 & 4 & -32 \\
-3 & 5 & -30 \\
-1 & 2 & -10 \\
\end{pmatrix}
\][/tex]
### Step 4: Calculate the determinants [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
To find [tex]\( D_x \)[/tex]:
[tex]\[
D_x = \det(D_x)
\][/tex]
[tex]\[
D_x = \begin{vmatrix}
-32 & 4 & -9 \\
-30 & 5 & -9 \\
-10 & 2 & -3
\end{vmatrix}
\][/tex]
Similarly, we calculate:
[tex]\[
D_y = \det(D_y)
\][/tex]
[tex]\[
D_y = \begin{vmatrix}
-2 & -32 & -9 \\
-3 & -30 & -9 \\
-1 & -10 & -3
\end{vmatrix}
\][/tex]
And finally:
[tex]\[
D_z = \det(D_z)
\][/tex]
[tex]\[
D_z = \begin{vmatrix}
-2 & 4 & -32 \\
-3 & 5 & -30 \\
-1 & 2 & -10
\end{vmatrix}
\][/tex]
### Step 5: Solve for [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] using Cramer's Rule
Cramer's Rule gives us the solution as:
[tex]\[
x = \frac{D_x}{D}
\][/tex]
[tex]\[
y = \frac{D_y}{D}
\][/tex]
[tex]\[
z = \frac{D_z}{D}
\][/tex]
Substituting the known determinant values:
[tex]\[
D = 3.0000000000000018
\][/tex]
[tex]\[
D_x = -5.999999999999996
\][/tex]
[tex]\[
D_y = 0.0
\][/tex]
[tex]\[
D_z = 12.000000000000005
\][/tex]
Therefore,
[tex]\[
x = \frac{D_x}{D} = \frac{-5.999999999999996}{3.0000000000000018} = -2
\][/tex]
[tex]\[
y = \frac{D_y}{D} = \frac{0.0}{3.0000000000000018} = 0
\][/tex]
[tex]\[
z = \frac{D_z}{D} = \frac{12.000000000000005}{3.0000000000000018} = 4
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = -2, \; y = 0, \; z = 4}
\][/tex]
