To solve the equation \(\frac{-6}{x+6} = \frac{-10}{x}\), we can use cross-multiplication. Here's the step-by-step solution:

1. **Cross-multiply** the fractions:
   \[
   -6 \cdot x = -10 \cdot (x + 6)
   \]

2. **Expand** the equation:
   \[
   -6x = -10x - 60
   \]

3. **Add** \(10x\) to both sides to get all the \(x\) terms on one side:
   \[
   4x = -60
   \]

4. **Divide** both sides by 4 to solve for \(x\):
   \[
   x = -15
   \]

Thus, the solution is \(x = -15\).

Would you like further details, or do you have any questions?

Here are some related questions to consider:
1. What happens if we substitute \(x = -15\) back into the original equation?
2. How would the solution change if the signs were different?
3. Can this equation have more than one solution? Why or why not?
4. How would you solve this equation if the terms were squared?
5. What if the equation was \(\frac{6}{x+6} = \frac{10}{x}\)?

**Tip:** Always check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the equation -6/(x+6) = -10/x with a detailed step-by-step solution. This problem involves solving equations with fractions and using cross-multiplication method. Suitable for students in Grades 7-9.

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