To solve this system of equations, we can follow these steps:
First, write the system of equations in matrix form [tex]\( AX = B \)[/tex], where [tex]\( A \)[/tex] is the coefficient matrix, [tex]\( X \)[/tex] is the vector of variables, and [tex]\( B \)[/tex] is the constants vector.
The system of equations is:
[tex]\[
\begin{array}{l}
-9x - 6z = 19 \\
y + 8z = -2 \\
x - 3y = 35
\end{array}
\][/tex]
We can represent it as:
[tex]\[
A = \begin{pmatrix}
-9 & 0 & -6 \\
0 & 1 & 8 \\
1 & -3 & 0
\end{pmatrix}
, \quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
, \quad
B = \begin{pmatrix}
19 \\
-2 \\
35
\end{pmatrix}
\][/tex]
To solve the system [tex]\(AX = B\)[/tex], we need to find the inverse of the matrix [tex]\(A\)[/tex] and multiply it by the matrix [tex]\(B\)[/tex], provided that [tex]\(A\)[/tex] is invertible. However, in this scenario, let's check whether there is a unique solution to this system by solving it directly.
Upon solving, we obtain the solution for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[
\left( x, y, z \right) = (-3.0, -12.666666666666666, 1.3333333333333333)
\][/tex]
Matching this solution with the options given:
Option a. [tex]\((0, -18, 16)\)[/tex]
Doesn't match, so incorrect.
Option b. [tex]\(\left( -\frac{19}{9}, 2, 0 \right)\)[/tex]
Doesn't match, so incorrect.
Option c. [tex]\(\left( -3, 2, \frac{4}{3} \right)\)[/tex]
Neither component [tex]\(y\)[/tex] nor [tex]\(z\)[/tex] matches the values calculated, so this is incorrect.
Option d. [tex]\(\left( -3, -\frac{38}{3}, \frac{4}{3} \right)\)[/tex]
[tex]\(x = -3\)[/tex] matches, but neither of [tex]\(y\)[/tex] nor [tex]\(z\)[/tex] match the values we obtained. Thus, this is also incorrect.
Upon further verification, none of the provided options in the multiple-choice exactly match the discovered solution. There might be a mismatch between the options and calculated solution, indicating a potential error in option presentation or the provided solution. Based on the calculated results, the unique solution set for the given system is:
[tex]\[
\left(x, y, z \right) = \left(-3, -\frac{38}{3}, \frac{4}{3}\right)
\][/tex]
If arrived at correctly through exact calculations (which our initial result doesn't quadratically align), the resolution or data might be inherent error needs adjustments in the provided choices for accuracy concerning linear algebra constraints compliance.
