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 using the formula [tex]\(\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1\)[/tex], we need to identify and plug in the appropriate values.
First, let's define the given values:
- The endpoints are [tex]\( Q = -14 \)[/tex] and [tex]\( S = 2 \)[/tex], so [tex]\( x_1 = -14 \)[/tex] and [tex]\( x_2 = 2 \)[/tex].
- The ratio is [tex]\( 3:5 \)[/tex], so [tex]\( m = 3 \)[/tex] and [tex]\( n = 5 \)[/tex].
Now, substitute these values into the formula:
[tex]\[
\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1
\][/tex]
Substituting the known values:
[tex]\[
\left(\frac{3}{3+5}\right)(2-(-14))+(-14)
\][/tex]
Simplify the fractions and the operation inside the parentheses:
[tex]\[
\left(\frac{3}{8}\right)(2+14)+(-14)
\][/tex]
This simplifies to exactly the correct expression:
[tex]\[
\left(\frac{3}{3+5}\right)(2-(-14))+(-14)
\][/tex]
Therefore, the expression that correctly uses the formula is:
[tex]\[
\boxed{\left(\frac{3}{3+5}\right)(2-(-14))+(-14)}
\][/tex]
This correctly partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a [tex]\( 3:5 \)[/tex] ratio and finds the location of point [tex]\( R \)[/tex]. Using this expression, the location of point [tex]\( R \)[/tex] on the number line is found to be at -8.0.
