We are given the equation:

$\frac{2x+1}{5} - \frac{3 - x}{6} = \frac{4 - 2x}{3}$

Step 1: Find the least common denominator (LCD)

The denominators are $5$, $6$, and $3$. The least common denominator (LCD) of these is 30.

Step 2: Multiply the entire equation by the LCD (30)

Multiply each term by 30 to eliminate the denominators:

$30 \cdot \left(\frac{2x+1}{5}\right) - 30 \cdot \left(\frac{3 - x}{6}\right) = 30 \cdot \left(\frac{4 - 2x}{3}\right)$

This simplifies to:

$6(2x + 1) - 5(3 - x) = 10(4 - 2x)$

Step 3: Distribute

Now distribute the constants to each term:

$6(2x + 1) = 12x + 6$ $-5(3 - x) = -15 + 5x$ $10(4 - 2x) = 40 - 20x$

So the equation becomes:

$12x + 6 - 15 + 5x = 40 - 20x$

Step 4: Simplify

Combine like terms:

$(12x + 5x) + (6 - 15) = 40 - 20x$ $17x - 9 = 40 - 20x$

Step 5: Move variables to one side

Add $20x$ to both sides:

$17x + 20x - 9 = 40$ $37x - 9 = 40$

Step 6: Isolate $x$

Add 9 to both sides:

$37x = 49$

Now divide by 37:

$x = \frac{49}{37}$

Thus, the solution is:

$x = \frac{49}{37}$

Would you like a detailed explanation or have any further questions?

Related Questions:

1. How do you solve equations with fractions more efficiently?
2. What is the process for solving linear equations with multiple fractions?
3. How do you determine the least common denominator?
4. What are other methods to isolate variables in an equation?
5. Can the solution method change for systems of linear equations with fractions?

Tip:

When dealing with equations that have multiple fractions, clearing the fractions by multiplying by the least common denominator can simplify the process considerably.