It seems like you're dealing with an equation involving fractions. Let me rewrite the problem and help solve it:

\[
\frac{1}{8 - y} = \frac{1}{8} + \frac{1}{y + 4}
\]

To solve this equation, we can follow these steps:

### Step 1: Combine the fractions on the right-hand side.
The right-hand side has two terms, \(\frac{1}{8}\) and \(\frac{1}{y+4}\). We need to find a common denominator for these fractions. The common denominator will be \(8(y + 4)\).

So, we can rewrite each fraction:
\[
\frac{1}{8} = \frac{y + 4}{8(y + 4)}, \quad \frac{1}{y + 4} = \frac{8}{8(y + 4)}
\]

Now, the right-hand side becomes:
\[
\frac{1}{8} + \frac{1}{y + 4} = \frac{y + 4 + 8}{8(y + 4)} = \frac{y + 12}{8(y + 4)}
\]

### Step 2: Rewrite the equation.
Now, the equation looks like this:
\[
\frac{1}{8 - y} = \frac{y + 12}{8(y + 4)}
\]

### Step 3: Cross-multiply.
We can now cross-multiply to eliminate the fractions:
\[
8(y + 4) = (8 - y)(y + 12)
\]

### Step 4: Expand both sides.
Let's expand both sides of the equation:

On the left-hand side:
\[
8(y + 4) = 8y + 32
\]

On the right-hand side:
\[
(8 - y)(y + 12) = 8y + 96 - y^2 - 12y = -y^2 - 4y + 96
\]

### Step 5: Set the equation to 0.
Now, let's set the equation equal to zero by bringing all terms to one side:
\[
8y + 32 = -y^2 - 4y + 96
\]
\[
0 = -y^2 - 4y + 96 - 8y - 32
\]
\[
0 = -y^2 - 12y + 64
\]

### Step 6: Solve the quadratic equation.
The equation is now:
\[
y^2 + 12y - 64 = 0
\]

We can solve this quadratic equation using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 12\), and \(c = -64\).

Substitute the values:
\[
y = \frac{-12 \pm \sqrt{12^2 - 4(1)(-64)}}{2(1)}
\]
\[
y = \frac{-12 \pm \sqrt{144 + 256}}{2}
\]
\[
y = \frac{-12 \pm \sqrt{400}}{2}
\]
\[
y = \frac{-12 \pm 20}{2}
\]

### Step 7: Find the solutions.
Now, solving for \(y\):
1. \(y = \frac{-12 + 20}{2} = \frac{8}{2} = 4\)
2. \(y = \frac{-12 - 20}{2} = \frac{-32}{2} = -16\)

### Final Answer:
The two possible solutions are:
\[
y = 4 \quad \text{or} \quad y = -16
\]

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you determine a common denominator for two fractions?
2. What are the steps involved in solving quadratic equations?
3. Can you explain how cross-multiplication helps in solving rational equations?
4. What does the discriminant in the quadratic formula represent?
5. How do you interpret the physical meaning of multiple solutions in real-world problems?

**Tip:** Always check for extraneous solutions when solving equations with fractions, especially if a variable could make a denominator zero!

Learn how to solve the rational equation 1/(8 - y) = 1/8 + 1/(y + 4) with a step-by-step solution involving combining fractions and solving a quadratic equation. This problem introduces key algebra concepts such as cross-multiplication and the quadratic formula. Ideal for students in Grades 9-11.

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1 8 − y

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