Let's solve the equation \( 9y^2z^2 + yz^2 - 9y = 0 \) step by step.

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

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

### Step 2: Solve each factor separately
We now have a product of two factors equal to zero. This implies two cases:

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

#### Case 1: \( y = 0 \)
If \( y = 0 \), this is one solution.

#### Case 2: Solve \( 9yz^2 + z^2 - 9 = 0 \)
Let's solve the quadratic equation \( 9yz^2 + z^2 - 9 = 0 \) for \( z \):

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

Now, divide both sides by \( 9y + 1 \) (assuming \( 9y + 1 \neq 0 \)):

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

Take the square root of both sides:

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

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

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

These are the solutions to the given equation.

Do you want further details or have any questions?

Here are five related questions to consider:

1. What would be the effect on the solution if the constant term \( -9 \) in the equation were different?
2. Can the expression \( z = \pm \frac{3}{\sqrt{9y + 1}} \) be simplified further for specific values of \( y \)?
3. How would you determine if there are any restrictions on \( y \) for real \( z \) values?
4. What is the nature of the roots for the quadratic equation \( 9yz^2 + z^2 - 9 = 0 \) in terms of discriminants?
5. How does factoring the common term \( y \) simplify the process of solving the equation?

**Tip:** Always check for the possibility of factoring in algebraic equations to simplify the solution process.

Learn how to solve the equation 9y^2z^2 + yz^2 - 9y = 0 with a detailed step-by-step solution. This problem involves factoring, solving quadratic equations, and exploring the relationship between variables y and z. Suitable for students in Grades 10-12.

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