4 x minus 8 y plus 4 z equals 5
minus 6 x plus 6 y minus 8 z equals 1
3 x plus 3 y plus 6 z equals negative 9

We are given the following system of linear equations:

\[
1. \ 4x - 8y + 4z = 5
\]
\[
2. \ -6x + 6y - 8z = 1
\]
\[
3. \ 3x + 3y + 6z = -9
\]

We can solve this system using methods such as substitution, elimination, or matrix operations. I'll proceed with the elimination method to find the values of \(x\), \(y\), and \(z\).

### Step 1: Simplify and eliminate variables

Let's start by simplifying and eliminating variables step by step.

#### Combine equations:
I'll first add and subtract equations to eliminate one of the variables. 

Let me calculate this for you.The system of equations has been reduced to the following expressions for \(x\) and \(y\) in terms of \(z\):

\[
x = -\frac{5z}{3} - \frac{19}{12}
\]
\[
y = -\frac{z}{3} - \frac{17}{12}
\]

This shows that \(x\) and \(y\) depend on \(z\). The system does not have a unique solution for specific values of \(x\), \(y\), and \(z\) but rather an infinite number of solutions where \(x\) and \(y\) vary as functions of \(z\).

Would you like further clarification on this, or would you like to proceed with finding specific values for \(x\), \(y\), and \(z\) based on an additional condition?

Here are some questions for further exploration:
1. What does it mean for a system of equations to have infinite solutions?
2. Can we find specific values for \(x\), \(y\), and \(z\) if another condition is added?
3. How can we graphically represent a system with infinite solutions?
4. Could we rewrite the system in matrix form and find a parametric solution?
5. What does elimination or substitution look like for systems with dependencies like this?

**Tip**: Systems with dependent equations often have solutions that lie on a plane or line in 3D space.

4x - 8y + 4z = 5, -6x + 6y - 8z = 1, 3x + 3y + 6z = -9

Explore how to solve the system of linear equations 4x - 8y + 4z = 5, -6x + 6y - 8z = 1, and 3x + 3y + 6z = -9 using elimination methods. This problem demonstrates concepts of linear algebra, including parametric solutions and dependent equations. Suitable for high school students in Grades 10-12.

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