Certainly! To solve for the location of point [tex]\( R \)[/tex] that partitions the directed line segment [tex]\( QS \)[/tex] in a [tex]\( 3:2 \)[/tex] ratio, we can use the section formula. The coordinates of [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] are given as [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] respectively.
Let's review the given values:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex]
- [tex]\( x_2 = 6 \)[/tex]
The section formula for a point [tex]\( R \)[/tex] that divides the segment joining points [tex]\( Q(x_1) \)[/tex] and [tex]\( S(x_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R = \frac{m x_2 + n x_1}{m + n} \][/tex]
Following this step-by-step:
1. Substitute the given values into the formula:
[tex]\[ R = \frac{3(6) + 2(-2)}{3 + 2} \][/tex]
2. Compute the numerator:
[tex]\[ 3(6) + 2(-2) = 18 - 4 = 14 \][/tex]
3. Compute the denominator:
[tex]\[ 3 + 2 = 5 \][/tex]
4. Divide the results of the numerator by the denominator to find [tex]\( R \)[/tex]:
[tex]\[ R = \frac{14}{5} = 2.8 \][/tex]
Therefore, the location of point [tex]\( R \)[/tex] on the number line is [tex]\( 2.8 \)[/tex]. This value matches one of the given fractions in the options:
[tex]\[ \frac{14}{5} \][/tex]
So, the location of point [tex]\( R \)[/tex] is [tex]\( \boxed{\frac{14}{5}} \)[/tex].
