https://file.aiquickdraw.com/mathbot/file/1724807027935h9tr9um0.jpg

The equation from the image is:

\[
\frac{2}{x + 8y} - \frac{1}{8x - y} = \frac{1}{x + 8y} + \frac{2}{8x - y}
\]

We need to find the values of \( x \) and \( y \).

### Step 1: Simplify the equation
First, let's bring all the terms to one side:

\[
\frac{2}{x + 8y} - \frac{1}{8x - y} - \frac{1}{x + 8y} - \frac{2}{8x - y} = 0
\]

Combine like terms:

\[
\frac{2 - 1}{x + 8y} - \frac{1 + 2}{8x - y} = 0
\]

This simplifies to:

\[
\frac{1}{x + 8y} - \frac{3}{8x - y} = 0
\]

### Step 2: Solve for \( x \) and \( y \)
To satisfy this equation, the two fractions must be equal:

\[
\frac{1}{x + 8y} = \frac{3}{8x - y}
\]

Cross-multiply to eliminate the fractions:

\[
8x - y = 3(x + 8y)
\]

### Step 3: Expand and collect like terms
Expand the right-hand side:

\[
8x - y = 3x + 24y
\]

Now, collect like terms:

\[
8x - 3x = 24y + y
\]

\[
5x = 25y
\]

### Step 4: Solve for \( x \) in terms of \( y \)
Divide both sides by 5:

\[
x = 5y
\]

### Step 5: Substitute back to find the relation
Using \( x = 5y \) in one of the original fractions:

\[
\frac{1}{x + 8y} = \frac{1}{5y + 8y} = \frac{1}{13y}
\]

Thus, \( 8x - y = 40y - y = 39y \) should satisfy \( \frac{3}{39y} = \frac{1}{13y} \).

So the value of \( y \) can be any non-zero number, and \( x \) is 5 times \( y \).

### Final Answer
The relationship between \( x \) and \( y \) is \( x = 5y \). The values of \( x \) and \( y \) must satisfy this relationship, and they can take any values that do not make the denominators zero.

Would you like further details, or have any questions?

### Related Questions:
1. What happens if \( y = 0 \) in the equation?
2. How would the solution change if the coefficients of the equation were different?
3. Can this method be used for other similar equations? If so, how?
4. How can you check if a solution satisfies the original equation?
5. What is the general approach to solving equations involving fractions?

### Tip:
When dealing with equations involving fractions, always consider cross-multiplying to eliminate the denominators, as it simplifies the process significantly.

Learn how to solve an equation involving fractions to find the values of x and y. This detailed step-by-step solution simplifies the equation, cross-multiplies to eliminate fractions, and establishes the relationship x = 5y. Suitable for advanced high school students and anyone interested in equations with fractions.

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