https://file.aiquickdraw.com/mathbot/file/1732236188979970o7oz8.jpg

To solve this, we calculate the weighted average of the points \( A(1, 3) \) and \( B(4, -6) \) using their weights \( w_A = 3 \) and \( w_B = 1 \).

### Formula for Weighted Average:
\[
P = \left( \frac{w_A \cdot x_A + w_B \cdot x_B}{w_A + w_B}, \frac{w_A \cdot y_A + w_B \cdot y_B}{w_A + w_B} \right)
\]

#### Substituting the given values:
- \( x_A = 1, y_A = 3 \)
- \( x_B = 4, y_B = -6 \)
- \( w_A = 3, w_B = 1 \)

1. Calculate the \( x \)-coordinate of \( P \):
\[
x_P = \frac{(3 \cdot 1) + (1 \cdot 4)}{3 + 1} = \frac{3 + 4}{4} = \frac{7}{4} = 1.75
\]

2. Calculate the \( y \)-coordinate of \( P \):
\[
y_P = \frac{(3 \cdot 3) + (1 \cdot -6)}{3 + 1} = \frac{9 - 6}{4} = \frac{3}{4} = 0.75
\]

### Coordinates of \( P \):
\[
P = (1.75, 0.75)
\]

Would you like additional details or further clarifications?

Here are some related questions:
1. What if the weights were equal for \( A \) and \( B \)?
2. How does changing the weights affect the position of \( P \)?
3. Can you explain how to calculate a weighted average in three dimensions?
4. What happens if one of the weights is zero?
5. Can you verify the solution geometrically on the graph?

**Tip**: When working with weighted averages, the result always lies within the range defined by the points weighted by their values.

Find the coordinates of P that represent the weighted average for the given set of points with the given weights. A has a weight of 3, and B has a weight of 1.

Learn how to calculate the weighted average of points A(1, 3) and B(4, -6) with weights 3 and 1. This step-by-step solution demonstrates how to apply the weighted average formula in coordinate geometry, suitable for Grades 8-10.

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