The line `y = 3x - 2` bisects the join of (a, 3) and (2, -5), find the value of a. 

Answer

To find the value of \( a \) such that the line \( y = 3x - 2 \) bisects the line segment joining the points \( (a, 3) \) and \( (2, -5) \), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( (a, 3) \) and \( (2, -5) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( (x_1, y_1) = (a, 3) \) and \( (x_2, y_2) = (2, -5) \).

Calculating the coordinates of the midpoint:
\[ M_x = \frac{a + 2}{2}, \quad M_y = \frac{3 + (-5)}{2} = \frac{3 - 5}{2} = \frac{-2}{2} = -1 \]
Thus, the midpoint \( M \) is:
\[ M = \left( \frac{a + 2}{2}, -1 \right) \]

### Step 2: Substitute the Midpoint into the Line Equation
Since the line \( y = 3x - 2 \) bisects the segment, it must pass through the midpoint \( M \). Therefore, we substitute \( M \) into the line equation:
\[ y = 3x - 2 \]
Substituting \( M_x \) and \( M_y \):
\[ -1 = 3\left(\frac{a + 2}{2}\right) - 2 \]

### Step 3: Solve for \( a \)
Now we solve the equation:
\[ -1 = \frac{3(a + 2)}{2} - 2 \]
Multiplying through by 2 to eliminate the fraction:
\[ -2 = 3(a + 2) - 4 \]
Simplifying the right side:
\[ -2 = 3a + 6 - 4 \]
\[ -2 = 3a + 2 \]
Subtracting 2 from both sides:
\[ -4 = 3a \]
Dividing by 3:
\[ a = -\frac{4}{3} \]

### Conclusion
The value of \( a \) is:
\[ \boxed{-\frac{4}{3}} \]