To solve the given system of linear equations for $x$, $y$, and $z$, we have:

1. $6x + y - 5z = -6$
2. $-9x - y + 7z = 5$
3. $-5x - y + 3z = -7$

Step 1: Eliminate $y$

Add the first and second equations to eliminate $y$: $(6x + y - 5z) + (-9x - y + 7z) = -6 + 5$ $-3x + 2z = -1 \quad \text{(Equation 4)}$

Add the first and third equations to eliminate $y$: $(6x + y - 5z) + (-5x - y + 3z) = -6 - 7$ $x - 2z = -13 \quad \text{(Equation 5)}$

Step 2: Solve the two-variable system

Now solve the two-variable system formed by Equation 4 and Equation 5:

1. $-3x + 2z = -1$ (Equation 4)
2. $x - 2z = -13$ (Equation 5)

From Equation 5: $x = -13 + 2z$

Substitute $x = -13 + 2z$ into Equation 4: $-3(-13 + 2z) + 2z = -1$ $39 - 6z + 2z = -1$ $39 - 4z = -1$ $-4z = -40 \quad \Rightarrow \quad z = 10$

Substitute $z = 10$ into $x = -13 + 2z$: $x = -13 + 2(10) = -13 + 20 = 7$

Step 3: Solve for $y$

Substitute $x = 7$ and $z = 10$ into the first equation: $6(7) + y - 5(10) = -6$ $42 + y - 50 = -6$ $y - 8 = -6$ $y = 2$

Final Solution:

$x = 7, \; y = 2, \; z = 10$

Let me know if you'd like detailed steps for verification or any further clarification.

Here are five follow-up questions you can explore:

1. What is the determinant of the coefficient matrix of this system?
2. Can this system be solved using matrix methods such as Gaussian elimination?
3. How would the solution change if one of the equations is modified?
4. What are some real-world applications of solving systems of equations like this?
5. Can this problem be solved graphically?

Tip: Always double-check your substitutions when solving simultaneous equations to avoid minor arithmetic errors.