Sure, let's solve the given system of linear equations step by step.
The given system of equations is:
[tex]\[
\left\{
\begin{array}{c}
x + 3y + 2z = 26 \\
x - 3y + 4z = 2 \\
2x + y + z = 8
\end{array}
\right.
\][/tex]
We need to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy all three equations simultaneously.
### Step 1: Write the system as a matrix
We can represent this system as a matrix equation [tex]\(AX = B\)[/tex]:
[tex]\[
A = \begin{bmatrix}
1 & 3 & 2 \\
1 & -3 & 4 \\
2 & 1 & 1
\end{bmatrix}
,
\quad
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
,
\quad
B = \begin{bmatrix}
26 \\
2 \\
8
\end{bmatrix}
\][/tex]
### Step 2: Solve the matrix equation [tex]\(AX = B\)[/tex]
To solve for [tex]\(X\)[/tex], we typically use the inverse of matrix [tex]\(A\)[/tex] if it exists, or other methods like row reduction or computational tools.
In this case, the solution to the system has been found and is:
[tex]\[
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
-1.42857143 \\
5.71428571 \\
5.14285714
\end{bmatrix}
\][/tex]
### Step 3: Convert the solution to fractions
We now convert the decimal solutions to fractions to match the provided answer choices.
[tex]\[
x = -1.42857143 = -\frac{10}{7}
\][/tex]
[tex]\[
y = 5.71428571 = \frac{40}{7}
\][/tex]
[tex]\[
z = 5.14285714 = \frac{36}{7}
\][/tex]
### Step 4: Select the correct answer from the provided choices
The calculated solution in fractional form is:
[tex]\[
\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)
\][/tex]
Comparing this with the given answer choices:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
B. [tex]\(\left( -\frac{10}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
C. [tex]\(\left( -\frac{5}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
D. [tex]\(\left( -\frac{5}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
The correct answer is:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
