https://file.aiquickdraw.com/mathbot/file/1728860453187m7yf00kw.jpg

The system of equations provided in the image is as follows:

\[
\begin{cases}
\frac{x - 2}{x + 2} = \frac{y - 7}{y - 5} \\
\frac{x + 1}{x - 1} = \frac{y - 3}{y - 5}
\end{cases}
\]

### Steps to solve the system of equations:
#### 1. Simplify the first equation:
\[
\frac{x - 2}{x + 2} = \frac{y - 7}{y - 5}
\]
Cross multiply:
\[
(x - 2)(y - 5) = (x + 2)(y - 7)
\]
Expanding both sides:
\[
xy - 5x - 2y + 10 = xy - 7x - 2y + 14
\]
Cancel terms and simplify:
\[
-5x + 10 = -7x + 14
\]
Solving for \(x\):
\[
2x = 4 \implies x = 2
\]

#### 2. Substitute \(x = 2\) into the second equation:
\[
\frac{x + 1}{x - 1} = \frac{y - 3}{y - 5}
\]
Substitute \(x = 2\):
\[
\frac{2 + 1}{2 - 1} = \frac{y - 3}{y - 5} \implies \frac{3}{1} = \frac{y - 3}{y - 5}
\]
Cross multiply:
\[
3(y - 5) = y - 3
\]
Expand and simplify:
\[
3y - 15 = y - 3
\]
Solving for \(y\):
\[
2y = 12 \implies y = 6
\]

Thus, the solution to the system is:
\[
x = 2, \quad y = 6
\]

Would you like a detailed explanation of any specific step or concept? Here are five related questions you might consider:

1. How do you approach solving systems of nonlinear equations?
2. What happens if a system of equations has no solution or infinitely many solutions?
3. Can you graphically interpret the solution to this system of equations?
4. What types of systems of equations require different methods (e.g., substitution, elimination)?
5. How would the solution change if a parameter were introduced into one of the equations?

**Tip:** When solving nonlinear systems, always check if simplifying or factoring expressions can reduce the complexity of the equations.

Solve the following system of equations:
\(\frac{x - 2}{x + 2} = \frac{y - 7}{y - 5}\)
\(\frac{x + 1}{x - 1} = \frac{y - 3}{y - 5}\)

This detailed explanation shows how to solve a system of rational equations step-by-step. The process includes cross-multiplying and simplifying expressions to find the values of x and y. Suitable for high school algebra students.

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