We are given the following equation:

\[
\frac{y - 9}{y - 2} = -8
\]

To solve for \( y \), let's break it down step-by-step:

### Step 1: Multiply both sides by \( (y - 2) \)
We do this to eliminate the denominator:

\[
y - 9 = -8 \times (y - 2)
\]

### Step 2: Expand the right-hand side
\[
y - 9 = -8y + 16
\]

### Step 3: Move all terms involving \( y \) to one side
Add \( 8y \) to both sides to group the \( y \)-terms together:

\[
y + 8y - 9 = 16
\]
\[
9y - 9 = 16
\]

### Step 4: Isolate the \( y \)-term
Add 9 to both sides:

\[
9y = 25
\]

### Step 5: Solve for \( y \)
Divide both sides by 9:

\[
y = \frac{25}{9}
\]

Thus, the solution is:

\[
y = \frac{25}{9}
\]

Let me know if you'd like further explanation on any step.

### 5 Related Questions:
1. What are the steps to solve a linear equation with variables on both sides?
2. How can you check if your solution to an equation is correct?
3. What happens if there’s no solution to a linear equation?
4. How do you solve equations involving fractions and variables?
5. How can you simplify expressions with parentheses before solving?

### Tip:
When solving equations with fractions, clear the fractions early by multiplying through by the denominator. This often simplifies the process.

Learn how to solve the linear equation y - 9 / y - 2 = -8 with a step-by-step solution. This problem involves core algebraic concepts like solving equations with variables on both sides and eliminating fractions through cross-multiplication. Suitable for students in Grades 8-10.

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