The expression you provided is:

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

Let's simplify it step by step.

1. First, distribute the \(3\) across the fraction:

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

2. Now add the constant \(1\):

\[
\frac{9y + 3}{2} + 1 = \frac{9y + 3}{2} + \frac{2}{2} = \frac{9y + 3 + 2}{2} = \frac{9y + 5}{2}
\]

So, the simplified form of the expression is:

\[
\frac{9y + 5}{2}
\]

Would you like more details or have any questions?

Here are some related questions for further practice:
1. Simplify: \( 4\left(\frac{2x + 3}{3}\right) + 2 \)
2. Solve for \(y\): \( 3\left(\frac{3y + 1}{2}\right) + 1 = 10 \)
3. Expand: \( 2(3y + 4) + 5 \)
4. Simplify: \( 5\left(\frac{2x + 1}{4}\right) + 3 \)
5. Solve for \(x\): \( \frac{5x + 3}{2} = 7 \)

**Tip**: When simplifying expressions with fractions, try to eliminate the fractions by multiplying through by the denominator or combining like terms carefully.

Learn how to simplify the algebraic expression 3[(3y + 1)/2] + 1. This detailed, step-by-step solution explains the distributive property and how to handle fractions in algebra. Perfect for students in Grades 7-8.

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