We are given the quadratic equation:

$by^2 - 3z = cy^2 + 5z - 7$

To solve for $y$, we need to first rearrange the terms so that all the terms involving $y$ are on one side of the equation. Let’s do that step by step:

Step 1: Move all terms involving $y$ to one side

We start by subtracting $cy^2$ from both sides:

$by^2 - cy^2 = 5z - 7 + 3z$

This simplifies to:

$(by^2 - cy^2) = 8z - 7$

Step 2: Factor out $y^2$

Now factor out $y^2$ from the left-hand side:

$y^2(b - c) = 8z - 7$

Step 3: Solve for $y^2$

Now divide both sides by $(b - c)$ to solve for $y^2$:

$y^2 = \frac{8z - 7}{b - c}$

Step 4: Solve for $y$

Finally, take the square root of both sides to solve for $y$:

$y = \sqrt{\frac{8z - 7}{b - c}}$

This is the solution for $y$.

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Do you have any questions or would you like further details? Here are some related questions to expand your understanding:

1. What happens if $b = c$ in this equation?
2. How would the equation change if we added a constant term on both sides?
3. How can you verify if this solution is valid for specific values of $b$, $c$, and $z$?
4. How does the solution change if there’s a different constant instead of $8z - 7$?
5. What is the impact of negative values for $z$ on the solution?

Tip: Always double-check if the denominator of a fraction could be zero; it could indicate a special case or undefined solution.