To solve the system of linear equations represented by the matrix equation:
[tex]\[
\begin{pmatrix}
1 & 3 & 1 \\
1 & -2 & -1 \\
2 & 1 & 2
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
10 \\
-6 \\
10
\end{pmatrix}
\][/tex]
we need to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]. Let's break it down step-by-step.
### Step 1: Write out the system of equations
From the given matrices, we can write the system of linear equations as follows:
1. [tex]\(1x + 3y + 1z = 10\)[/tex]
2. [tex]\(1x - 2y - 1z = -6\)[/tex]
3. [tex]\(2x + 1y + 2z = 10\)[/tex]
### Step 2: Use methods to solve the system
There are various methods to solve such a system (substitution, elimination, matrix methods, etc.). Here, we'll perform the solution typically found through matrix methods without delving into extensive manual calculation details.
### Step 3: Perform matrix operations
We proceed by finding the inverse of the coefficient matrix and then multiplying it by the constant vector. Practically, we solve the equation [tex]\(A \mathbf{x} = \mathbf{b}\)[/tex] by calculating [tex]\(\mathbf{x} = A^{-1} \mathbf{b}\)[/tex].
### Step 4: Interpret the solution
The solution gives us the values for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[
(x, y, z) = (1.0, 2.0, 3.0)
\][/tex]
### Solution:
Upon solving this system, we find that:
[tex]\[
x = 1.0, \quad y = 2.0, \quad z = 3.0
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 1, \quad y = 2, \quad z = 3
\][/tex]
This solution satisfies all three original equations:
1. [tex]\(1(1) + 3(2) + 1(3) = 1 + 6 + 3 = 10\)[/tex]
2. [tex]\(1(1) - 2(2) - 1(3) = 1 - 4 - 3 = -6\)[/tex]
3. [tex]\(2(1) + 1(2) + 2(3) = 2 + 2 + 6 = 10\)[/tex]
Thus, the solution is verified to be correct.
