To solve the given system of linear equations, we need to determine the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously. The system of equations is:
1. [tex]\( x - y + z = -3 \)[/tex]
2. [tex]\( 2x + y - 3z = 3 \)[/tex]
3. [tex]\( 8x - 3y + z = -9 \)[/tex]
Let’s solve this step-by-step.
### Step 1: Write the System of Equations in Matrix Form
We can represent the system of equations as a matrix equation [tex]\( A\mathbf{x} = \mathbf{b} \)[/tex], where:
[tex]\[
A = \begin{pmatrix}
1 & -1 & 1 \\
2 & 1 & -3 \\
8 & -3 & 1
\end{pmatrix}
\][/tex]
[tex]\[
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\][/tex]
[tex]\[
\mathbf{b} = \begin{pmatrix}
-3 \\
3 \\
-9
\end{pmatrix}
\][/tex]
### Step 2: Row Reduction or Applying Inverses
To solve the matrix equation [tex]\( A\mathbf{x} = \mathbf{b} \)[/tex], we can use techniques such as row reduction (Gaussian elimination) or matrix inversion. However, this solution assumes correctly solving the equations to find the unique solution.
### Step 3: Find the Solution
After performing the appropriate calculations (details of eliminating variables and obtaining the solution are assumed to be done correctly):
The solution obtained is:
[tex]\[
\mathbf{x} = \begin{pmatrix}
4.44 \times 10^{-16} \\
3.00 \\
7.77 \times 10^{-16}
\end{pmatrix}
\][/tex]
### Step 4: Interpret the Solution
The entries [tex]\( 4.44 \times 10^{-16} \)[/tex] and [tex]\( 7.77 \times 10^{-16} \)[/tex] are very close to zero, indicating that those values can be considered as zero due to numerical precision errors commonly encountered in calculations.
Therefore, the solution to the system of equations is:
[tex]\[
x \approx 0, \quad y = 3, \quad z \approx 0
\][/tex]
### Final Solution
Hence, the values of the variables that satisfy the system of equations are approximately:
[tex]\[
x = 0, \quad y = 3, \quad z = 0
\][/tex]
