The points A(-3, 2), B(2, -1) and C(a, 4) are collinear. Find a. 

Answer

To find the value of \( a \) such that the points \( A(-3, 2) \), \( B(2, -1) \), and \( C(a, 4) \) are collinear, we can use the concept of slopes. The points are collinear if the slope of line segment \( AB \) is equal to the slope of line segment \( BC \).

### Step-by-Step Solution:

1. **Find the slope of line segment \( AB \)**:
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( A(-3, 2) \) and \( B(2, -1) \):
- \( x_1 = -3, y_1 = 2 \)
- \( x_2 = 2, y_2 = -1 \)

Plugging in the values:
\[ \text{slope of } AB = \frac{-1 - 2}{2 - (-3)} = \frac{-3}{5} \]

2. **Find the slope of line segment \( BC \)**:
For points \( B(2, -1) \) and \( C(a, 4) \):
- \( x_1 = 2, y_1 = -1 \)
- \( x_2 = a, y_2 = 4 \)

Using the slope formula:
\[ \text{slope of } BC = \frac{4 - (-1)}{a - 2} = \frac{5}{a - 2} \]

3. **Set the slopes equal to each other**:
Since the points are collinear, we set the slopes equal:
\[ \frac{-3}{5} = \frac{5}{a - 2} \]

4. **Cross-multiply to solve for \( a \)**:
Cross-multiplying gives:
\[ -3(a - 2) = 5 \cdot 5 \]
Simplifying this:
\[ -3a + 6 = 25 \]

5. **Rearrange the equation**:
\[ -3a = 25 - 6 \]
\[ -3a = 19 \]

6. **Solve for \( a \)**:
Dividing both sides by -3:
\[ a = -\frac{19}{3} \]

### Final Answer:
The value of \( a \) is \( -\frac{19}{3} \).