To solve the equation \( 9y^2z^2 + yz^2 - 9y = 0 \), let's proceed by factoring the expression.

### Step 1: Factor out the common terms
First, notice that each term in the equation has a common factor of \( y \). We can factor \( y \) out:

\[
y(9yz^2 + z^2 - 9) = 0
\]

### Step 2: Solve the factored equation
Now, we have a product of two factors equal to zero. Therefore, we set each factor equal to zero:

1. \( y = 0 \)
2. \( 9yz^2 + z^2 - 9 = 0 \)

#### Solution 1:
\( y = 0 \)

This is one solution.

#### Solution 2:
Solve the quadratic equation \( 9yz^2 + z^2 - 9 = 0 \).

This can be rewritten as:

\[
z^2(9y + 1) - 9 = 0
\]

Now, move \( -9 \) to the other side:

\[
z^2(9y + 1) = 9
\]

Divide both sides by \( 9y + 1 \) (assuming \( 9y + 1 \neq 0 \)):

\[
z^2 = \frac{9}{9y + 1}
\]

Take the square root of both sides:

\[
z = \pm \sqrt{\frac{9}{9y + 1}} = \pm \frac{3}{\sqrt{9y + 1}}
\]

### Final Solutions:
The solutions to the equation are:

1. \( y = 0 \)
2. \( z = \pm \frac{3}{\sqrt{9y + 1}} \)

Let me know if you have any questions or need further details!

Here are five related questions to consider:

1. What happens if \( y = -\frac{1}{9} \) in the original equation?
2. How would you graph the solutions in the \( y \)-\( z \) plane?
3. How can the discriminant be used to analyze the solutions for different values of \( y \)?
4. How would the solutions change if the coefficient of \( yz^2 \) were different?
5. What are the real-world applications of solving such quadratic equations?

**Tip:** When solving quadratic equations, always check if you can factor out common terms first to simplify the equation.

Explore the detailed solution to the equation 9y^2z^2 + yz^2 - 9y = 0. Learn how to factorize and solve quadratic equations involving multiple variables. Suitable for advanced high school students and learners interested in algebraic problem-solving.

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