To solve the equation \(\frac{z+1}{z} - \frac{8}{7} = 0\), let's follow these steps:

### Step 1: Eliminate the fraction
First, move \(\frac{8}{7}\) to the other side of the equation:
\[
\frac{z+1}{z} = \frac{8}{7}
\]

### Step 2: Multiply both sides by \(z\)
Multiplying both sides of the equation by \(z\) to eliminate the denominator on the left-hand side:
\[
z+1 = \frac{8z}{7}
\]

### Step 3: Eliminate the fraction on the right-hand side
Multiply the entire equation by 7 to eliminate the fraction:
\[
7(z + 1) = 8z
\]
which simplifies to:
\[
7z + 7 = 8z
\]

### Step 4: Solve for \(z\)
Subtract \(7z\) from both sides:
\[
7 = z
\]

### Final Solution:
The solution to the equation is:
\[
z = 7
\]

Would you like further details on any step or have questions? Here are five related questions for further practice:

1. Solve \(\frac{2z+3}{z} - \frac{5}{6} = 0\).
2. What happens if you solve \(\frac{z-4}{z} - \frac{9}{7} = 0\)?
3. Can you solve \(\frac{3z+5}{2z} - \frac{4}{5} = 0\)?
4. Solve \(\frac{z+2}{z-1} = \frac{3}{4}\).
5. What is the solution for \(\frac{z+6}{2z+3} - \frac{7}{5} = 0\)?

**Tip:** Always clear fractions early by multiplying through by the denominator. It simplifies the equation.

This guide explains how to solve the rational equation Z+1/z - 8/7 = 0 with clear step-by-step instructions. Learn how to manipulate rational expressions and use cross-multiplication to simplify and solve. Suitable for students in Grades 8-10.

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