We have the following system of three equations with three variables:

1. $x - 5z = y$
2. $2x = 3y + z - 6$
3. $4y - 2z = 7 - 7x$

Step 1: Express $y$ from the first equation

From equation (1):
$x - 5z = y$ We can solve for $y$: $y = x - 5z$

Step 2: Substitute $y = x - 5z$ into the other two equations

Substitution into equation (2):

$2x = 3y + z - 6$ Substitute $y = x - 5z$: $2x = 3(x - 5z) + z - 6$ Simplify: $2x = 3x - 15z + z - 6$ $2x = 3x - 14z - 6$ Subtract $3x$ from both sides: $- x = -14z - 6$ Multiply by $-1$: $x = 14z + 6$

Step 3: Substitute $x = 14z + 6$ into $y = x - 5z$

From equation (1):
$y = x - 5z$ Substitute $x = 14z + 6$: $y = (14z + 6) - 5z$ Simplify: $y = 9z + 6$

Step 4: Substitute $x = 14z + 6$ and $y = 9z + 6$ into equation (3)

The third equation is: $4y - 2z = 7 - 7x$ Substitute $y = 9z + 6$ and $x = 14z + 6$: $4(9z + 6) - 2z = 7 - 7(14z + 6)$ Simplify both sides: $36z + 24 - 2z = 7 - 98z - 42$ $34z + 24 = -35 - 98z$ Move all terms involving $z$ to one side: $34z + 98z = -35 - 24$ $132z = -59$ Solve for $z$: $z = -\frac{59}{132}$

Step 5: Solve for $x$ and $y$

Now, substitute $z = -\frac{59}{132}$ into $x = 14z + 6$ and $y = 9z + 6$:

Solve for $x$:

$x = 14\left(-\frac{59}{132}\right) + 6$ Simplify: $x = -\frac{826}{132} + 6 = -\frac{826}{132} + \frac{792}{132} = -\frac{34}{132} = -\frac{17}{66}$

Solve for $y$:

$y = 9\left(-\frac{59}{132}\right) + 6$ Simplify: $y = -\frac{531}{132} + 6 = -\frac{531}{132} + \frac{792}{132} = \frac{261}{132} = \frac{87}{44}$

Final Solution:

$x = -\frac{17}{66}, \quad y = \frac{87}{44}, \quad z = -\frac{59}{132}$

Would you like any details or have any questions about the steps?

Here are 5 related questions to deepen your understanding:

1. How would the solution change if the constant terms in each equation were different?
2. Can you solve the system using matrix methods, such as Gaussian elimination?
3. How can this solution be verified by substitution into the original equations?
4. What geometric interpretation can be derived from solving this system of equations?
5. How would the solution change if the system was dependent or inconsistent?

Tip: When solving systems of equations, always try to simplify one equation to express one variable, then substitute it into others to reduce complexity.