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The given equation is:

$-\frac{2}{1} = -\frac{8}{7} - \frac{4}{1}z$

First, simplify $-\frac{2}{1}$ and $-\frac{4}{1}$ to $-2$ and $-4$ respectively:

$-2 = -\frac{8}{7} - 4z$

Next, add $\frac{8}{7}$ to both sides to isolate the term with $z$:

$-2 + \frac{8}{7} = -4z$

Convert $-2$ to a fraction with a denominator of 7:

$-\frac{14}{7} + \frac{8}{7} = -4z$

Combine the fractions:

$-\frac{6}{7} = -4z$

Now, divide both sides by $-4$ to solve for $z$:

$z = \frac{\frac{6}{7}}{4}$

Simplify the expression:

$z = \frac{6}{28} = \frac{3}{14}$

Thus, the solution is:

$z = \frac{3}{14}$

Would you like more details or have any questions?

Here are five related questions to expand on this concept:

1. How would the solution change if the equation was $-\frac{2}{1} = -\frac{8}{7} + \frac{4}{1}z$?
2. Can you solve for $z$ if the equation was $2 = -\frac{8}{7} - 4z$?
3. What happens if the equation was multiplied by 7 on both sides?
4. How would you solve the equation if the term with $z$ had a different denominator?
5. What are the implications if $z$ is required to be a whole number?

Tip: When solving equations, always ensure all terms are simplified first, and fractions have a common denominator if necessary.