https://file.aiquickdraw.com/mathbot/file/1727312451120yrjkyl03.jpg

The equation shown in the image is:

\[
9 = -y^3 + 1
\]

To solve for \( y \), we can follow these steps:

### Step 1: Isolate the \( y^3 \) term
\[
9 = -y^3 + 1
\]
Subtract 1 from both sides:
\[
9 - 1 = -y^3
\]
\[
8 = -y^3
\]

### Step 2: Solve for \( y^3 \)
Multiply both sides by -1:
\[
-8 = y^3
\]

### Step 3: Solve for \( y \)
Take the cube root of both sides:
\[
y = \sqrt[3]{-8}
\]
\[
y = -2
\]

So, the solution is \( y = -2 \).

Let me know if you'd like more details or have any questions!

Here are five related questions to extend your understanding:
1. What happens if the right-hand side of the equation is \( y^3 - 1 \) instead?
2. How do we solve \( y^3 = 8 \)?
3. What is the general method for solving cubic equations?
4. How do you solve equations with higher powers of \( y \), such as \( y^4 \)?
5. What is the cube root of a complex number?

**Tip**: When dealing with cube roots of negative numbers, always remember that real numbers have a unique real cube root, unlike square roots, which may not be real.

Learn how to solve the cubic equation 9 = -y^3 + 1 with a detailed step-by-step solution. This problem covers key algebraic concepts, including solving cubic equations and applying cube roots. Suitable for students in Grades 9-12.

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