To solve this problem, we need to find a point that is two-fifths of the way from point $A(-3, 5)$ to point $B(3, 2)$ using an algebraic method.

Step-by-step Solution:

We use the section formula to find the point $P$ that divides the line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a given ratio. The formula for the point dividing the segment in the ratio $m:n$ is given by:

$P(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)$

In this case, $A(x_1, y_1) = (-3, 5)$ and $B(x_2, y_2) = (3, 2)$, and we are told the point is two-fifths of the way from $A$ to $B$. This means the ratio is 2:3 (since two-fifths implies that the remaining part is three-fifths).

Thus, $m = 2$ and $n = 3$.

Now, applying the section formula:

$P(x, y) = \left( \frac{2 \cdot 3 + 3 \cdot (-3)}{2 + 3}, \frac{2 \cdot 2 + 3 \cdot 5}{2 + 3} \right)$

Calculation:

For the $x$-coordinate:

$x = \frac{2 \cdot 3 + 3 \cdot (-3)}{5} = \frac{6 - 9}{5} = \frac{-3}{5} = -0.6$

For the $y$-coordinate:

$y = \frac{2 \cdot 2 + 3 \cdot 5}{5} = \frac{4 + 15}{5} = \frac{19}{5} = 3.8$

Final Answer:

The point that is two-fifths of the way from $A(-3, 5)$ to $B(3, 2)$ is:

$P(-0.6, 3.8)$

Let me know if you need further clarification or details.

Here are 5 related questions to consider:

1. How would the formula change if the ratio was three-fifths instead of two-fifths?
2. Can you generalize the section formula for any ratio?
3. What would the midpoint of the segment between $A(-3, 5)$ and $B(3, 2)$ be?
4. How does the concept of dividing a line segment relate to vector algebra?
5. What are other methods to find a point on a line segment besides the section formula?

Tip: The section formula is particularly useful for dividing line segments in both 2D and 3D geometry. Understanding it can also help in problems related to ratios and proportions in coordinate geometry.