To solve the given system of equations:
[tex]\[
\begin{cases}
6x + 12y - 6z = 6 \\
18x - 24y + 18z = -6 \\
-42x + 36y - 30z = 36
\end{cases}
\][/tex]
we'll go through the process of solving a system of linear equations using standard algebraic methods, specifically elimination or matrix methods. However, after analyzing the equations carefully, we can determine the following:
1. Start with the first equation:
[tex]\[
6x + 12y - 6z = 6
\][/tex]
2. For the second equation:
[tex]\[
18x - 24y + 18z = -6
\][/tex]
Let's divide the entire second equation by 6 to simplify it:
[tex]\[
3x - 4y + 3z = -1
\][/tex]
3. For the third equation:
[tex]\[
-42x + 36y - 30z = 36
\][/tex]
Let's divide the entire third equation by -6 to simplify it:
[tex]\[
7x - 6y + 5z = -6
\][/tex]
Now, let's compare the simplified equations:
1. [tex]\(6x + 12y - 6z = 6\)[/tex]
2. [tex]\(3x - 4y + 3z = -1\)[/tex]
3. [tex]\(7x - 6y + 5z = -6\)[/tex]
We can observe possible relationships or dependencies between the equations. Nonetheless, let's attempt to solve the system.
### Analyze the Equations
To check for consistency, let's convert the system to an augmented matrix form and use Gaussian elimination or similar methods:
[tex]\[
\begin{pmatrix}
6 & 12 & -6 & | & 6 \\
3 & -4 & 3 & | & -1 \\
7 & -6 & 5 & | & -6
\end{pmatrix}
\][/tex]
#### Begin Row Operations:
Step 1: Simplify the first row:
Divide the first row by 6:
[tex]\[
\begin{pmatrix}
1 & 2 & -1 & | & 1 \\
3 & -4 & 3 & | & -1 \\
7 & -6 & 5 & | & -6
\end{pmatrix}
\][/tex]
Step 2: Eliminate [tex]\(x\)[/tex] from the second and third rows:
Subtract [tex]\(3 \times\)[/tex] the first row from the second row:
[tex]\[
\begin{pmatrix}
1 & 2 & -1 & | & 1 \\
0 & -10 & 6 & | & -4 \\
7 & -6 & 5 & | & -6
\end{pmatrix}
\][/tex]
Subtract [tex]\(7 \times\)[/tex] the first row from the third row:
[tex]\[
\begin{pmatrix}
1 & 2 & -1 & | & 1 \\
0 & -10 & 6 & | & -4 \\
0 & -20 & 12 & | & -13
\end{pmatrix}
\][/tex]
Step 3: Simplify the second and third rows:
Divide the second row by -10:
[tex]\[
\begin{pmatrix}
1 & 2 & -1 & | & 1 \\
0 & 1 & -0.6 & | & 0.4 \\
0 & -20 & 12 & | & -13
\end{pmatrix}
\][/tex]
Step 4: Eliminate [tex]\(y\)[/tex] from the third row:
Add [tex]\(20 \times\)[/tex] the second row to the third row:
[tex]\[
\begin{pmatrix}
1 & 2 & -1 & | & 1 \\
0 & 1 & -0.6 & | & 0.4 \\
0 & 0 & 0 & | & -5
\end{pmatrix}
\][/tex]
The appearance of the row [tex]\([0, 0, 0 | -5]\)[/tex] in the augmented matrix indicates that the system of equations is inconsistent; no solution exists that can satisfy all three equations simultaneously.
Conclusion:
Given the appearance of inconsistent values in our analysis, the system of equations is inconsistent, meaning that there is no set of [tex]\((x, y, z)\)[/tex] that satisfies all given equations. Thus, the system does not have any solution.
