To solve the equation $-\frac{3}{5} + \frac{7}{6}x = \frac{7}{3}x$, follow these steps:

Step 1: Eliminate the fraction by finding a common denominator.

First, let's eliminate the fractions by finding the least common multiple (LCM) of the denominators. The denominators are 5, 6, and 3. The LCM of 5, 6, and 3 is 30.

Step 2: Multiply the entire equation by 30.

$30 \times \left(-\frac{3}{5}\right) + 30 \times \left(\frac{7}{6}x\right) = 30 \times \left(\frac{7}{3}x\right)$

This simplifies to: $-18 + 35x = 70x$

Step 3: Solve for $x$.

To isolate $x$, subtract $35x$ from both sides: $-18 = 35x - 70x$

Combine like terms: $-18 = -35x$

Finally, divide both sides by $-35$: $x = \frac{-18}{-35}$

Simplify the fraction: $x = \frac{18}{35}$

So, the solution is $x = \frac{18}{35}$.

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

Here are some related questions to explore:

1. How do you solve linear equations with fractions?
2. What is the method to find the least common multiple (LCM) of several numbers?
3. How do you check the solution to an equation by substitution?
4. What are the steps to simplify fractions?
5. How does the distributive property help in solving equations?

Tip: When solving equations with fractions, multiplying the entire equation by the least common multiple of the denominators can simplify the process significantly.