https://file.aiquickdraw.com/mathbot/file/1725763295132p5gzxvuz.jpg

The problem presented in the image is as follows:

\[
\frac{1}{u} + \frac{2}{5} = \frac{7}{u} - 25
\]

### Steps to solve for \( u \):

1. **Move all terms with \( u \) to one side:**
   We subtract \(\frac{7}{u}\) from both sides:

   \[
   \frac{1}{u} - \frac{7}{u} + \frac{2}{5} = -25
   \]

   Simplifying the terms with \( u \):

   \[
   \frac{1 - 7}{u} + \frac{2}{5} = -25
   \]

   \[
   \frac{-6}{u} + \frac{2}{5} = -25
   \]

2. **Isolate the term with \( u \):**
   Subtract \(\frac{2}{5}\) from both sides:

   \[
   \frac{-6}{u} = -25 - \frac{2}{5}
   \]

   To combine these, express \(-25\) as a fraction:

   \[
   \frac{-6}{u} = \frac{-125}{5} - \frac{2}{5}
   \]

   \[
   \frac{-6}{u} = \frac{-127}{5}
   \]

3. **Solve for \( u \):**
   Cross-multiply to eliminate the fractions:

   \[
   -6 \cdot 5 = u \cdot (-127)
   \]

   \[
   -30 = -127u
   \]

   Divide by \(-127\):

   \[
   u = \frac{30}{127}
   \]

Thus, the solution is:

\[
u = \frac{30}{127}
\]

Would you like more details on any step? Here are some relative questions for further exploration:

1. How do we handle equations with multiple fractions like this?
2. Could \( u = 0 \) be a solution in rational equations?
3. What if the denominators were different? How would that affect the solution process?
4. How does cross-multiplication work in rational equations?
5. What happens when there are no solutions to an equation like this?

**Tip:** When solving rational equations, it's helpful to eliminate the fractions by finding a common denominator or cross-multiplying directly!

Learn how to solve rational equations with a detailed step-by-step solution. This problem involves algebraic concepts such as cross-multiplication and isolating terms. Suitable for students in Grades 8-10.

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