The formula you're using is the section formula for finding a point that divides a line segment in a given ratio. The formula is:

$\text{Location of point } R = \left(\frac{m}{m + n}\right)(x_2 - x_1) + x_1$

Where:

• $x_1$ and $x_2$ are the coordinates of the endpoints of the directed line segment (Q and S, respectively).
• $m$ and $n$ are the ratio parts that partition the segment (in your case, $m = 3$ and $n = 5$).

Given:

• Q = -14 (i.e., $x_1 = -14$)
• S = 2 (i.e., $x_2 = 2$)
• The ratio $m:n = 3:5$

Applying the formula:

$\text{Location of point R} = \left(\frac{3}{3 + 5}\right)(2 - (-14)) + (-14)$

Simplify this:

$= \left(\frac{3}{8}\right)(2 + 14) + (-14)$ $= \left(\frac{3}{8}\right)(16) - 14$ $= 6 - 14 = -8$

Thus, point R is located at -8.

The correct expression from your options is:

$\left(\frac{3}{3 + 5}\right)(2 - (-14)) + (-14)$

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Would you like any additional details or further clarification on this solution?

Additional Questions:

1. How do you apply the section formula in cases with different ratios?
2. What does the formula tell us about partitioning a directed line segment?
3. How can you apply this formula to a 2D coordinate plane?
4. How does the section formula change if the ratio is negative?
5. Can you use this formula to find midpoints of line segments?

Tip: Make sure to simplify expressions step by step. It can help to separate the addition and subtraction inside the parentheses before multiplying.