To solve the equation \( \frac{1}{2}x - \frac{5}{6} = \frac{3}{7}x - \frac{1}{3} \), follow these steps:

1. **Eliminate the fractions** by finding the least common denominator (LCD). The LCD for 2, 6, 7, and 3 is 42. Multiply every term by 42 to eliminate the fractions:

   \[
   42 \times \left(\frac{1}{2}x\right) - 42 \times \left(\frac{5}{6}\right) = 42 \times \left(\frac{3}{7}x\right) - 42 \times \left(\frac{1}{3}\right)
   \]

2. **Simplify each term**:

   \[
   21x - 35 = 18x - 14
   \]

3. **Isolate the \(x\) terms** by subtracting \(18x\) from both sides:

   \[
   21x - 18x - 35 = -14
   \]
   \[
   3x - 35 = -14
   \]

4. **Add 35 to both sides** to isolate the \(x\) term:

   \[
   3x = 21
   \]

5. **Divide both sides by 3** to solve for \(x\):

   \[
   x = 7
   \]

The solution is:
\[
x = 7
\]

Would you like further details or have any questions?

Here are 8 related questions to expand on this topic:

1. How do you solve linear equations with variables on both sides?
2. What is the least common denominator, and how do you find it?
3. How can multiplying every term by the LCD help in solving equations?
4. How do you solve an equation that has both positive and negative terms?
5. What is the process for checking if the solution to an equation is correct?
6. How do you handle equations that contain fractions and variables?
7. What are some strategies to simplify equations before solving them?
8. How do you solve systems of linear equations?

**Tip:** When dealing with equations involving fractions, eliminating the fractions by multiplying through by the least common denominator can simplify the process significantly.

Learn how to solve the equation 1/2x - 5/6 = 3/7x - 1/3 with detailed steps and explanations. This problem involves equations with fractions and variable isolation techniques. Suitable for students in Grades 7-9.

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