https://file.aiquickdraw.com/mathbot/file/1726857875945fg60a3di.jpg

The equation shown in the image is:

\[
\frac{x + 10}{5x - 10} = \frac{x}{5}
\]

### Steps to simplify and solve for \(x\):

1. **Cross-multiply** to eliminate the denominators:
   \[
   (x + 10) \cdot 5 = x \cdot (5x - 10)
   \]

   This results in:
   \[
   5(x + 10) = x(5x - 10)
   \]

2. **Distribute** on both sides:
   \[
   5x + 50 = 5x^2 - 10x
   \]

3. **Move all terms to one side** to form a quadratic equation:
   \[
   0 = 5x^2 - 10x - 5x - 50
   \]

   Simplifying:
   \[
   5x^2 - 15x - 50 = 0
   \]

4. **Divide through by 5** to simplify the equation:
   \[
   x^2 - 3x - 10 = 0
   \]

5. **Solve the quadratic equation** using the quadratic formula:
   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}
   \]
   \[
   x = \frac{3 \pm \sqrt{9 + 40}}{2}
   \]
   \[
   x = \frac{3 \pm \sqrt{49}}{2}
   \]
   \[
   x = \frac{3 \pm 7}{2}
   \]

6. **Find the two solutions**:
   \[
   x = \frac{3 + 7}{2} = 5
   \]
   \[
   x = \frac{3 - 7}{2} = -2
   \]

### The two possible solutions are \(x = 5\) and \(x = -2\).

However, we need to **check for extraneous solutions** by substituting these values back into the original equation to ensure they do not make the denominator zero.

- For \(x = 5\):
  \[
  5x - 10 = 5(5) - 10 = 25 - 10 = 15 \quad (\text{not zero, so } x = 5 \text{ is valid})
  \]

- For \(x = -2\):
  \[
  5x - 10 = 5(-2) - 10 = -10 - 10 = -20 \quad (\text{not zero, so } x = -2 \text{ is also valid})
  \]

### Final Answer:
The solutions are \(x = 5\) and \(x = -2\).

Would you like a deeper explanation of any step? Here are 5 follow-up questions:

1. How can we derive the quadratic formula?
2. What are extraneous solutions, and why do we check for them?
3. Can you solve a similar equation with different coefficients?
4. How does cross-multiplying help solve rational equations?
5. What are other methods for solving quadratic equations?

**Tip**: Always check for restrictions by ensuring that values don't make the denominator zero!

Simplify and solve the equation: (x + 10) / (5x - 10) = x / 5

Learn how to simplify and solve the rational equation (x + 10) / (5x - 10) = x / 5. This step-by-step guide covers cross-multiplication and solving a quadratic equation using the quadratic formula. Suitable for Grades 10-11 students.

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