Sure, let’s solve the system of linear equations using Cramer's Rule. The system of equations given is:
[tex]\[
\begin{cases}
-x + 2y - 3z = 1 \\
2x + z = 0 \\
3x - 4y + 4z = 2
\end{cases}
\][/tex]
First, let's write the system in matrix form [tex]\( AX = B \)[/tex], where [tex]\( A \)[/tex] is the coefficient matrix, [tex]\( X \)[/tex] is the column vector of variables, and [tex]\( B \)[/tex] is the column vector of constants.
[tex]\[
A = \begin{pmatrix}
-1 & 2 & -3 \\
2 & 0 & 1 \\
3 & -4 & 4
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
B = \begin{pmatrix}
1 \\
0 \\
2
\end{pmatrix}
\][/tex]
### Step 1: Calculate Determinant of [tex]\( A \)[/tex]
The determinant of [tex]\( A \)[/tex] is denoted as [tex]\( \text{det}(A) \)[/tex].
[tex]\[
\text{det}(A) = 10.000000000000002
\][/tex]
Since [tex]\(\text{det}(A) \neq 0\)[/tex], the system has a unique solution.
### Step 2: Calculate Determinant of [tex]\( A_1 \)[/tex]
Matrix [tex]\( A_1 \)[/tex] is formed by replacing the first column of [tex]\( A \)[/tex] with the column vector [tex]\( B \)[/tex]:
[tex]\[
A_1 = \begin{pmatrix}
1 & 2 & -3 \\
0 & 0 & 1 \\
2 & -4 & 4
\end{pmatrix}
\][/tex]
The determinant of [tex]\( A_1 \)[/tex] is [tex]\(\text{det}(A_1) = 7.999999999999998\)[/tex].
### Step 3: Calculate Determinant of [tex]\( A_2 \)[/tex]
Matrix [tex]\( A_2 \)[/tex] is formed by replacing the second column of [tex]\( A \)[/tex] with the column vector [tex]\( B \)[/tex]:
[tex]\[
A_2 = \begin{pmatrix}
-1 & 1 & -3 \\
2 & 0 & 1 \\
3 & 2 & 4
\end{pmatrix}
\][/tex]
The determinant of [tex]\( A_2 \)[/tex] is [tex]\(\text{det}(A_2) = -15.000000000000007\)[/tex].
### Step 4: Calculate Determinant of [tex]\( A_3 \)[/tex]
Matrix [tex]\( A_3 \)[/tex] is formed by replacing the third column of [tex]\( A \)[/tex] with the column vector [tex]\( B \)[/tex]:
[tex]\[
A_3 = \begin{pmatrix}
-1 & 2 & 1 \\
2 & 0 & 0 \\
3 & -4 & 2
\end{pmatrix}
\][/tex]
The determinant of [tex]\( A_3 \)[/tex] is [tex]\(\text{det}(A_3) = -16.000000000000007\)[/tex].
### Step 5: Solve for [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] using Cramer's Rule
Cramer's Rule states that the solution for the variables [tex]\( x, y, z \)[/tex] can be found using:
[tex]\[
x = \frac{\text{det}(A_1)}{\text{det}(A)} = \frac{7.999999999999998}{10.000000000000002} = 0.7999999999999997
\][/tex]
[tex]\[
y = \frac{\text{det}(A_2)}{\text{det}(A)} = \frac{-15.000000000000007}{10.000000000000002} = -1.5000000000000004
\][/tex]
[tex]\[
z = \frac{\text{det}(A_3)}{\text{det}(A)} = \frac{-16.000000000000007}{10.000000000000002} = -1.6000000000000005
\][/tex]
### Final Solution
The solution to the system of equations is:
[tex]\[
\boxed{x = 0.7999999999999997, \, y = -1.5000000000000004, \, z = -1.6000000000000005}
\][/tex]
