https://file.aiquickdraw.com/mathbot/file/1725401757141iwkvtibp.jpg

The equation shown in the image is:

\[
\frac{6 + 5x}{3x} = \frac{7}{x}
\]

To solve this equation for \(x\), let's follow these steps:

1. **Cross multiply** to eliminate the fractions:

\[
(6 + 5x) \cdot x = 7 \cdot 3x
\]

2. **Expand** both sides:

\[
6x + 5x^2 = 21x
\]

3. **Rearrange** the equation to set it to zero:

\[
5x^2 + 6x - 21x = 0
\]

\[
5x^2 - 15x = 0
\]

4. **Factor** the quadratic equation:

\[
5x(x - 3) = 0
\]

5. **Solve** for \(x\):

This equation will be true if either \(5x = 0\) or \(x - 3 = 0\).

- \(5x = 0\) gives \(x = 0\)
- \(x - 3 = 0\) gives \(x = 3\)

However, \(x = 0\) is not a valid solution because it would make the original equation undefined (division by zero). So, the only solution is:

\[
x = 3
\]

Would you like any more details or have any questions?

Here are some related questions to explore:

1. How do you solve other quadratic equations like \(ax^2 + bx + c = 0\)?
2. What are the conditions under which a quadratic equation has real solutions?
3. How does cross-multiplying help in solving rational equations?
4. What is the importance of checking for extraneous solutions in equations?
5. Can you provide examples of other equations where solutions are restricted?

**Tip:** Always check the solutions in the original equation, especially when dealing with rational expressions, to ensure they don't lead to division by zero or other undefined operations.

Solve the equation (6 + 5x) / (3x) = 7 / x.

Learn how to solve the rational equation (6 + 5x) / (3x) = 7 / x with a detailed step-by-step solution. Understand key algebraic concepts, including cross-multiplication and solving quadratic equations. Suitable for students in Grades 8-10.

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