*by*2−3*z* = *cy*2+5*z*−7, for y **(don't put the plus or minus (±) here as the system will reject it** ) *y* =

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 \( y \).

---
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.

Learn how to solve a quadratic equation involving multiple variables: by^2 - 3z = cy^2 + 5z - 7. This problem explores algebraic concepts like factoring and using square roots to solve for y, with a step-by-step explanation. Suitable for Grades 9-11.

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