Given the augmented matrix below, solve the associated system of equations. For your variables, use x1, x2, x3, x4, x5, and x6. [[1, -6, -5, -6, 3, 0 ,-1],[0,0,0,0,1,-5,0],[0,0,0,0,0,1,-9]] the solution is (_,_,_,_,_,_)

We are given the following augmented matrix:

\[
\left[\begin{array}{ccccccc}
1 & -6 & -5 & -6 & 3 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & -5 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & -9 \\
\end{array}\right]
\]

The goal is to solve the associated system of linear equations. We will express the solution in terms of the variables \( x_1, x_2, x_3, x_4, x_5, x_6 \).

### Step 1: Convert the augmented matrix to a system of equations

From the augmented matrix, we can write down the system of equations:

1. \( x_1 - 6x_2 - 5x_3 - 6x_4 + 3x_5 = -1 \)
2. \( x_5 - 5x_6 = 0 \)
3. \( x_6 = 9 \)

### Step 2: Solve the system

From equation (3), we already know:
\[
x_6 = 9
\]

Substitute \( x_6 = 9 \) into equation (2):
\[
x_5 - 5(9) = 0 \implies x_5 = 45
\]

Now substitute \( x_5 = 45 \) and \( x_6 = 9 \) into equation (1):
\[
x_1 - 6x_2 - 5x_3 - 6x_4 + 3(45) = -1
\]
Simplify:
\[
x_1 - 6x_2 - 5x_3 - 6x_4 + 135 = -1
\]
\[
x_1 - 6x_2 - 5x_3 - 6x_4 = -136
\]

### Step 3: General solution

We can express the solution in terms of the free variables \( x_2, x_3, x_4 \), since we have one equation with 4 unknowns. Let \( x_2 = t_1 \), \( x_3 = t_2 \), and \( x_4 = t_3 \), where \( t_1, t_2, t_3 \) are parameters.

Thus, we get:
\[
x_1 = -136 + 6t_1 + 5t_2 + 6t_3
\]

### Final solution:
\[
(x_1, x_2, x_3, x_4, x_5, x_6) = (-136 + 6t_1 + 5t_2 + 6t_3, t_1, t_2, t_3, 45, 9)
\]

This is the parametric solution to the system of equations, where \( t_1, t_2, t_3 \) are free parameters.

Would you like further details or clarifications?

Here are five related questions for further exploration:
1. What are free variables in a system of linear equations?
2. How does Gaussian elimination help solve systems of equations?
3. What does it mean for a system to have infinitely many solutions?
4. How do augmented matrices represent systems of equations?
5. How can we verify a solution to a system of linear equations?

**Tip**: When solving systems of equations with more unknowns than equations, it's common to express solutions in terms of free variables.

Given the augmented matrix below, solve the associated system of equations. For your variables, use x1, x2, x3, x4, x5, and x6. [[1, -6, -5, -6, 3, 0 ,-1],[0,0,0,0,1,-5,0],[0,0,0,0,0,1,-9]] the solution is (_,_,_,_,_,_).

Learn how to solve a system of equations using an augmented matrix with free variables. This problem demonstrates the Gaussian elimination method and explains how to express solutions in parametric form. Suitable for advanced high school or college-level students.

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