To determine the solution to the given system of equations, we need to analyze the system for consistency and solvability. Let's rewrite the system for clarity:
[tex]\[
\begin{array}{c}
2x - y + 3z = 8 \\
x - 6y - z = 0 \\
-6x + 3y - 9z = 24
\end{array}
\][/tex]
Let's analyze this system step-by-step.
### Step 1: Represent the System in Matrix Form
First, write the coefficients in an augmented matrix form:
[tex]\[
\begin{bmatrix}
2 & -1 & 3 & | & 8 \\
1 & -6 & -1 & | & 0 \\
-6 & 3 & -9 & | & 24
\end{bmatrix}
\][/tex]
### Step 2: Simplify Using Row Operations
We shall row reduce the matrix to find if it is consistent.
1. Swap rows if necessary: The first row is already in a suitable form.
2. Eliminate [tex]\(x\)[/tex] from the second and third rows:
[tex]\[ \text{R2} \leftarrow \text{R2} - \frac{1}{2} \text{R1} \Rightarrow (1 - \frac{1}{2}(2))x + (-6 + \frac{1}{2}(1))y - (1 + \frac{1}{2}(3))z = 0 - \frac{1}{2}(8) \][/tex]
[tex]\[ \Rightarrow 0x + (-6.5)y + (-2.5)z = -4 \][/tex]
[tex]\[ \Rightarrow \begin{bmatrix}
2 & -1 & 3 & | & 8 \\
0 & -6.5 & -2.5 & | & -4 \\
-6 & 3 & -9 & | & 24
\end{bmatrix} \][/tex]
[tex]\[ \text{R3} \leftarrow \text{R3} + 3 \cdot \text{R1} \Rightarrow (-6 + 6)x + (3 - 3)y + (-9 + 9)z = 24 + 24 \][/tex]
[tex]\[ \Rightarrow 0x + 0y + 0z = 48 \][/tex]
[tex]\[ \Rightarrow \begin{bmatrix}
2 & -1 & 3 & | & 8 \\
0 & -6.5 & -2.5 & | & -4 \\
0 & 0 & 0 & | & 48
\end{bmatrix} \][/tex]
### Step 3: Analyze the Result
From the matrix, we can see the last row translates to [tex]\(0 = 48\)[/tex]. This is a contradiction since there is no value of [tex]\(x\)[/tex], [tex]\(y\)[/tex], or [tex]\(z\)[/tex] that can satisfy this equation.
Therefore, the system of equations is inconsistent and has no solution. This corresponds to answer:
C. No solution
