The given equation is:

$\frac{y - 3}{4} - \frac{2y + 4}{3} = 1 - \frac{y + 7}{2}$

Let me solve this step by step.

Step 1: Eliminate the denominators

To eliminate the fractions, we need to find the least common denominator (LCD) of 4, 3, and 2. The LCD is 12. Multiply every term in the equation by 12:

$12 \times \left( \frac{y - 3}{4} \right) - 12 \times \left( \frac{2y + 4}{3} \right) = 12 \times 1 - 12 \times \left( \frac{y + 7}{2} \right)$

This simplifies to:

$3(y - 3) - 4(2y + 4) = 12 - 6(y + 7)$

Step 2: Simplify the expressions

Expand each term:

$3y - 9 - (8y + 16) = 12 - 6y - 42$

Simplify further:

$3y - 9 - 8y - 16 = 12 - 6y - 42$

$-5y - 25 = -6y - 30$

Step 3: Solve for $y$

Add $6y$ to both sides:

$y - 25 = -30$

Now add 25 to both sides:

$y = -5$

Final Answer:

The solution to the equation is $y = -5$.

Would you like more details on any of these steps or have further questions?

Here are 5 related questions for practice:

1. Solve: $\frac{x + 1}{3} - \frac{x - 2}{4} = \frac{5}{6}$
2. Simplify: $\frac{3y - 7}{5} + \frac{2y + 4}{3} = \frac{10}{15}$
3. Solve: $2x - \frac{x}{3} = \frac{5x}{6}$
4. What is the least common denominator of 8, 6, and 4?
5. How do you handle an equation with variables on both sides?

Tip: When solving equations with fractions, eliminating the denominators early by multiplying through by the least common denominator can simplify the process significantly.