Let's solve the equation step-by-step:

The equation is:

\[
-2y + 3 = -\frac{4}{9}y - \frac{2}{3} - \frac{1}{3}y
\]

### Step 1: Combine like terms on the right side.

First, let's combine the \(y\) terms on the right side:

\[
-\frac{4}{9}y - \frac{1}{3}y = -\frac{4}{9}y - \frac{3}{9}y = -\frac{7}{9}y
\]

Now the equation is:

\[
-2y + 3 = -\frac{7}{9}y - \frac{2}{3}
\]

### Step 2: Clear the fractions by finding a common denominator.

The least common multiple of 9 and 3 is 9. Let's multiply every term in the equation by 9 to eliminate the fractions:

\[
9(-2y) + 9 \cdot 3 = 9\left(-\frac{7}{9}y\right) + 9\left(-\frac{2}{3}\right)
\]

This simplifies to:

\[
-18y + 27 = -7y - 6
\]

### Step 3: Move all terms involving \(y\) to one side and constants to the other.

Add \(7y\) to both sides to get all the \(y\) terms on one side:

\[
-18y + 7y + 27 = -6
\]

Simplify:

\[
-11y + 27 = -6
\]

Now, subtract 27 from both sides to get all the constants on the other side:

\[
-11y = -6 - 27
\]

Simplify:

\[
-11y = -33
\]

### Step 4: Solve for \(y\).

Divide both sides by \(-11\):

\[
y = \frac{-33}{-11}
\]

Simplify:

\[
y = 3
\]

### Solution:

\[
y = 3
\]

Would you like more details or have any questions?

Here are some related questions you might consider:
1. What would the solution be if the equation were \(-2y + 3 = -\frac{5}{9}y - \frac{2}{3} - \frac{1}{3}y\)?
2. How would the solution change if one of the fractions was positive, such as \(-2y + 3 = \frac{4}{9}y - \frac{2}{3} - \frac{1}{3}y\)?
3. What are the steps to solve an equation with mixed numbers instead of fractions?
4. How do you check if your solution to an equation is correct?
5. What techniques can you use if there are variables in both the numerator and the denominator of fractions?

**Tip:** When solving equations with fractions, consider multiplying through by the least common multiple to simplify the equation and make it easier to solve.

Learn how to solve the equation -2y + 3 = -4/9y - 2/3 - 1/3y with a detailed step-by-step solution. This problem involves equations with fractions and requires combining like terms and solving for a variable. Suitable for students in Grades 7-9.

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