To solve the given system of equations using an inverse matrix, follow these detailed steps:
### Step 1: Write the system of equations in matrix form.
The system of equations given is:
[tex]\[
\begin{array}{l}
x + 5y - 3z = -10 \\
-5x + 6y - 5z = -21 \\
-x + 8y - 8z = -25
\end{array}
\][/tex]
This can be written as a matrix equation [tex]\(A \mathbf{x} = \mathbf{B}\)[/tex], where:
[tex]\[
A = \begin{pmatrix}
1 & 5 & -3 \\
-5 & 6 & -5 \\
-1 & 8 & -8
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
\mathbf{B} = \begin{pmatrix}
-10 \\
-21 \\
-25
\end{pmatrix}
\][/tex]
### Step 2: Check if the matrix [tex]\(A\)[/tex] is invertible.
To find the solution, we need to check if the matrix [tex]\(A\)[/tex] is invertible. A matrix is invertible if its determinant is non-zero.
(The check has shown that [tex]\(A\)[/tex] is invertible.)
### Step 3: Find the inverse of matrix [tex]\(A\)[/tex].
Since [tex]\(A\)[/tex] is invertible, we can find the inverse [tex]\(A^{-1}\)[/tex]. The solution to the system of equations is given by:
[tex]\[
\mathbf{x} = A^{-1} \mathbf{B}
\][/tex]
### Step 4: Perform the matrix multiplication.
Using the inverse of the matrix [tex]\(A\)[/tex] and the vector [tex]\(\mathbf{B}\)[/tex], we perform the multiplication to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
The calculation yields:
[tex]\[
\mathbf{x} = \begin{pmatrix}
1 \\
-1 \\
2
\end{pmatrix}
\][/tex]
So the solution to the system of equations is:
[tex]\[
x = 1, \quad y = -1, \quad z = 2
\][/tex]
### Step 5: Compare the solution with the given choices.
The given choices were:
1. [tex]\([1, -1, 2]\)[/tex]
2. [tex]\([-6, -5, 2]\)[/tex]
3. [tex]\([-6, -2, -2]\)[/tex]
4. No solution
Comparing the solution [tex]\((1, -1, 2)\)[/tex] with the given choices, we find that it matches choice 1.
### Conclusion:
The solution to the system of equations is [tex]\((1, -1, 2)\)[/tex], which corresponds to choice 1. Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
