To solve the equation

\[
\frac{1}{x} = \frac{6}{5x} + 1,
\]

follow these steps:

1. **Simplify the equation:**

   Start by getting rid of the fractions. Multiply every term by \(x\) to eliminate the denominators:

   \[
   1 = \frac{6}{5} + x.
   \]

2. **Multiply by 5 to clear fractions:**

   To remove the fraction, multiply everything by 5:

   \[
   5 = 6 + 5x.
   \]

3. **Isolate the variable \(x\):**

   Now, solve for \(x\) by isolating it on one side of the equation:

   \[
   5x = 5 - 6,
   \]

   Simplify further:

   \[
   5x = -1.
   \]

4. **Divide by 5:**

   Finally, divide both sides by 5:

   \[
   x = -\frac{1}{5}.
   \]

So, the solution is \( x = -\frac{1}{5} \).

Would you like further details or have any questions?

Here are some related questions that might interest you:

1. How do you solve equations involving fractions?
2. What are the steps to clear fractions in an equation?
3. How do you solve equations with multiple variables?
4. What happens if the variable is in the denominator on both sides?
5. How do you check the solution of a rational equation?

**Tip:** When solving equations with fractions, multiply through by the least common multiple of the denominators to simplify the process.

Learn how to solve the equation 1/x = 6/(5x) + 1 with a detailed step-by-step solution. This problem involves equations and fractions, focusing on eliminating fractions and isolating variables. Suitable for students in Grades 7-9.

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