Add the following algebraic expressions and then find value at y =1
`(2y)/(3) - (5y^(2))/(3) + (5y^(3))/(2), - (4)/(3) + (2y^(2))/(3) - (y)/(2)`

Options

A. `(1)/(3)`
B. `(-3)/(7)`
C. `(3)/(2)`
D. `(4)/(3)`

Answer

To solve the problem, we need to add the given algebraic expressions and then find the value at \( y = 1 \).

### Step 1: Write down the expressions
We have two algebraic expressions:
1. \( \frac{2y}{3} - \frac{5y^2}{3} + \frac{5y^3}{2} \)
2. \( -\frac{4}{3} + \frac{2y^2}{3} - \frac{y}{2} \)

### Step 2: Combine the expressions
We can combine these two expressions into one:
\[ \left(\frac{2y}{3} - \frac{5y^2}{3} + \frac{5y^3}{2}\right) + \left(-\frac{4}{3} + \frac{2y^2}{3} - \frac{y}{2}\right) \]

### Step 3: Group like terms
Now, we will group the like terms together:
- Terms with \( y^3 \): \( \frac{5y^3}{2} \)
- Terms with \( y^2 \): \( -\frac{5y^2}{3} + \frac{2y^2}{3} \)
- Terms with \( y \): \( \frac{2y}{3} - \frac{y}{2} \)
- Constant terms: \( -\frac{4}{3} \)

### Step 4: Simplify each group
1. **For \( y^2 \)**:
\[ -\frac{5y^2}{3} + \frac{2y^2}{3} = -\frac{3y^2}{3} = -y^2 \]

2. **For \( y \)**:
To add \( \frac{2y}{3} - \frac{y}{2} \), we need a common denominator, which is 6:
\[ \frac{2y}{3} = \frac{4y}{6}, \quad -\frac{y}{2} = -\frac{3y}{6} \]
Therefore:
\[ \frac{4y}{6} - \frac{3y}{6} = \frac{1y}{6} = \frac{y}{6} \]

3. **Constant term**:
The constant term remains as \( -\frac{4}{3} \).

### Step 5: Combine all simplified terms
Putting it all together, we have:
\[ \frac{5y^3}{2} - y^2 + \frac{y}{6} - \frac{4}{3} \]

### Step 6: Substitute \( y = 1 \)
Now, we substitute \( y = 1 \) into the expression:
\[ \frac{5(1)^3}{2} - (1)^2 + \frac{1}{6} - \frac{4}{3} \]
This simplifies to:
\[ \frac{5}{2} - 1 + \frac{1}{6} - \frac{4}{3} \]

### Step 7: Find a common denominator
The common denominator for \( 2, 1, 6, \) and \( 3 \) is \( 6 \):
- \( \frac{5}{2} = \frac{15}{6} \)
- \( -1 = -\frac{6}{6} \)
- \( \frac{1}{6} = \frac{1}{6} \)
- \( -\frac{4}{3} = -\frac{8}{6} \)

### Step 8: Combine the fractions
Now we can combine:
\[ \frac{15}{6} - \frac{6}{6} + \frac{1}{6} - \frac{8}{6} = \frac{15 - 6 + 1 - 8}{6} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{3} \]