To determine which expression correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1\)[/tex] to find the location of point [tex]\(R\)[/tex] that partitions the directed line segment from [tex]\(Q\)[/tex] to [tex]\(S\)[/tex] in a [tex]\(3:5\)[/tex] ratio, let's examine each option carefully.
Given:
- [tex]\(Q = -14\)[/tex]
- [tex]\(S = 2\)[/tex]
- [tex]\(m = 3\)[/tex]
- [tex]\(n = 5\)[/tex]
The formula for finding the point [tex]\(R\)[/tex] that partitions the segment in the ratio [tex]\(m:n\)[/tex] is:
[tex]\[R = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1\][/tex]
### Step-by-step Analysis:
#### Option 1:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(2 + 14) + (-14)
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(16) + (-14)
\][/tex]
[tex]\[
6 - 14 = -8
\][/tex]
This simplifies to [tex]\(-8\)[/tex].
#### Option 2:
[tex]\[
\left(\frac{3}{3+5}\right)(-14 - 2) + 2
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(-14 - 2) + 2
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(-16) + 2
\][/tex]
[tex]\[
-6 + 2 = -4
\][/tex]
This simplifies to [tex]\(-4\)[/tex].
#### Option 3:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - 14) + 14
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(2 - 14) + 14
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(-12) + 14
\][/tex]
[tex]\[
-4.5 + 14 = 9.5
\][/tex]
This simplifies to [tex]\(9.5\)[/tex].
#### Option 4:
[tex]\[
\left(\frac{3}{3+5}\right)(-14 - 2) - 2
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(-14 - 2) - 2
\][/tex]
[tex]\[
\left(\frac{3}{8}\right)(-16) - 2
\][/tex]
[tex]\[
-6 - 2 = -8
\][/tex]
This simplifies to [tex]\(-8\)[/tex].
By comparing these calculations, we see that the correct expression is the one that simplifies to [tex]\(-8\)[/tex], which is the right position for point [tex]\(R\)[/tex].
Thus, the correct expression is:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
This is Option 1.
