https://file.aiquickdraw.com/mathbot/file/1726132532999lsdfttnx.jpg

Let's examine and solve the systems of equations provided in the image.

### a)
\[
\begin{aligned}
3x + 2y + z &= 2 \\
4x + 2y + 2z &= 8 \\
x - y + z &= 4
\end{aligned}
\]
We will solve this system step by step.

### b)
\[
\begin{aligned}
2x + 4y + 6z &= -12 \\
2x - 3y - 4z &= 15 \\
3x + 4y + 5z &= -8
\end{aligned}
\]
We'll solve this system and check for consistency.

### c)
\[
\begin{aligned}
x + 4y - z &= 12 \\
3x + 8y - 2z &= 4
\end{aligned}
\]
This system contains two equations and three variables, so we'll check if there is a consistent solution.

### d)
\[
\begin{aligned}
x + y + 3z &= 12 \\
2x + 2y + 6z &= 6
\end{aligned}
\]
We'll determine if this system has a solution or if it's inconsistent.

### e)
\[
\begin{aligned}
x + y + z &= 1 \\
2x + y + 2z &= 2 \\
3x + 2y + 3z &= 5
\end{aligned}
\]
We'll solve this to check for consistency.

I'll now start solving these systems of equations. Let's begin with the first system (a).For system **(a)**, the solution is:
\[
x = -4, \quad y = 2, \quad z = 10
\]
This system is **consistent**, and we found a unique solution.

Now, let's proceed to solve system **(b)**.It seems like I can't do more advanced data analysis right now. Please try again later. If you'd like, you can resubmit this request and I'll attempt it again.

Would you like to continue with some other parts of the problem manually, or shall I help with something else?

Here are some related questions you might find interesting:
1. What are the criteria for determining if a system of linear equations is consistent or inconsistent?
2. How can we interpret the solution of a system of linear equations geometrically?
3. What methods are available to solve a system of linear equations apart from substitution and elimination?
4. How does the number of variables and equations affect the solution of a system of linear equations?
5. What are dependent, independent, and inconsistent systems in linear algebra?

**Tip**: When solving a system of equations, always check the determinant for square systems. A zero determinant implies either no solution or infinitely many solutions.

Examine the consistency of the following system of equations and solve if possible.

Explore how to examine the consistency of systems of linear equations through methods such as Gaussian elimination and substitution. This problem set contains multiple systems that vary in complexity, making it perfect for students learning about linear algebra and methods of solving equations.

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