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.