https://file.aiquickdraw.com/mathbot/file/1725974630077xxlmfz8e.jpg

The equation in the image is:

\[
\frac{x}{x^2 + x + 1} = \frac{y}{y^2 + y + 1} \implies xy = 1 \quad \text{or} \quad x = y
\]

### Detailed Steps to Solve:

1. The given equation is:

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

2. To eliminate the denominators, we cross-multiply:

\[
x(y^2 + y + 1) = y(x^2 + x + 1)
\]

3. Expanding both sides:

\[
x(y^2 + y + 1) = xy^2 + xy + x
\]
\[
y(x^2 + x + 1) = yx^2 + yx + y
\]

4. Simplifying the expanded form:

\[
xy^2 + xy + x = yx^2 + yx + y
\]

5. Cancel the common terms \( xy \) from both sides:

\[
xy^2 + x = yx^2 + y
\]

6. Rearrange to group terms:

\[
xy^2 - yx^2 = y - x
\]

This is a complex equation to solve directly, but the right side of the image gives a hint that either:

- \( xy = 1 \), or
- \( x = y \).

### Verification of Solutions:

1. **If \( x = y \):**

   Substitute \( x = y \) into the original equation:

   \[
   \frac{x}{x^2 + x + 1} = \frac{x}{x^2 + x + 1}
   \]

   This is clearly true, so \( x = y \) is a valid solution.

2. **If \( xy = 1 \):**

   Substitute \( y = \frac{1}{x} \) into the original equation:

   \[
   \frac{x}{x^2 + x + 1} = \frac{\frac{1}{x}}{\frac{1}{x^2} + \frac{1}{x} + 1}
   \]

   Simplifying the right-hand side:

   \[
   \frac{\frac{1}{x}}{\frac{1 + x + x^2}{x^2}} = \frac{1}{x^2 + x + 1}
   \]

   This also holds true, so \( xy = 1 \) is another valid solution.

### Final Conclusion:
The solutions are:
- \( x = y \), or
- \( xy = 1 \).

---

Do you have any questions or want more details on any step?

Here are some related questions you could explore:
1. How can you verify the solution \( xy = 1 \) graphically?
2. What does the solution \( x = y \) represent geometrically?
3. How would the solution change if the equation was modified slightly?
4. What if the equation was \( \frac{x}{x^2 + x + 2} = \frac{y}{y^2 + y + 2} \)?
5. What happens when \( x \) and \( y \) are complex numbers?

**Tip**: Cross-multiplying is a powerful technique for eliminating denominators when solving rational equations.

Solve the equation \(\frac{x}{x^2 + x + 1} = \frac{y}{y^2 + y + 1}\) and determine the relationship between x and y.

Learn how to solve the rational equation \(\frac{x}{x^2 + x + 1} = \frac{y}{y^2 + y + 1}\) through cross-multiplication. This algebra problem covers key concepts like rational equations and offers two solutions: x = y or xy = 1. Suitable for students in Grades 10-12.

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