https://file.aiquickdraw.com/mathbot/file/1726628207867l0po8h2g.jpg

The equation in the image is:

\[
\frac{3(y + 3)}{y + 1} + 2 = \frac{3y + 1}{y + 1}
\]

Let's solve this step by step.

### Step 1: Simplify the left-hand side
The first term is \(\frac{3(y + 3)}{y + 1}\), which can be expanded as:

\[
\frac{3(y + 3)}{y + 1} = \frac{3y + 9}{y + 1}
\]

Now, rewrite the equation:

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

### Step 2: Combine like terms
To combine terms, we will express \(2\) in terms of a fraction with a denominator of \(y + 1\):

\[
2 = \frac{2(y + 1)}{y + 1} = \frac{2y + 2}{y + 1}
\]

Now the equation becomes:

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

### Step 3: Combine the fractions on the left-hand side
Since the denominators are the same, we can combine the numerators:

\[
\frac{(3y + 9) + (2y + 2)}{y + 1} = \frac{3y + 1}{y + 1}
\]

Simplifying the numerator:

\[
\frac{5y + 11}{y + 1} = \frac{3y + 1}{y + 1}
\]

### Step 4: Eliminate the denominator
Since the denominators on both sides are equal, we can set the numerators equal to each other:

\[
5y + 11 = 3y + 1
\]

### Step 5: Solve for \(y\)
Subtract \(3y\) from both sides:

\[
2y + 11 = 1
\]

Subtract 11 from both sides:

\[
2y = -10
\]

Finally, divide by 2:

\[
y = -5
\]

### Final Answer:
The solution to the equation is \(y = -5\).

Would you like any further details or have questions? Here are some related questions to expand on:

1. How do you handle fractions with different denominators in an equation?
2. Can equations with variables in the denominator have extraneous solutions?
3. What are the implications of canceling denominators in algebraic fractions?
4. How would this problem change if the denominators were not identical?
5. How does this method relate to solving equations with rational expressions?

**Tip:** Always check for potential restrictions on the variable when dealing with denominators to avoid division by zero.

Solve the equation: (3(y + 3))/(y + 1) + 2 = (3y + 1)/(y + 1)

Learn how to solve rational equations like (3(y + 3))/(y + 1) + 2 = (3y + 1)/(y + 1) with step-by-step solutions. This problem covers key algebraic concepts and techniques for solving linear equations with rational expressions, suitable for students in Grades 9-11.

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