Let's solve the given equation step by step:

The equation is:

\[
\frac{0.1z - 0.9}{0.2z} = 0
\]

### Step 1: Eliminate the denominator by multiplying both sides by \(0.2z\):

\[
0.1z - 0.9 = 0
\]

### Step 2: Solve for \(z\)

Now, isolate \(z\):

\[
0.1z = 0.9
\]

### Step 3: Divide both sides by 0.1:

\[
z = \frac{0.9}{0.1} = 9
\]

So, the solution is:

\[
z = 9
\]

Would you like more details on how this was solved or have any further questions?

Here are some related questions you might consider:
1. How do you solve equations with variables in both the numerator and denominator?
2. What happens if \(z = 0\) in this equation?
3. Can this equation be solved graphically?
4. How do decimals affect the accuracy of solving equations?
5. What if the equation had a quadratic term instead?

**Tip:** Always check for any restrictions in equations involving denominators to avoid division by zero!

Learn how to solve the linear equation 0.1z - 0.9 = 0.2z step-by-step. This problem introduces algebraic manipulation, isolating variables, and solving equations with variables in both numerator and denominator. Suitable for students in Grades 6-8.

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