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On a number line, the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] has endpoints [tex]\( Q \)[/tex] at [tex]\(-2\)[/tex] and [tex]\( S \)[/tex] at [tex]\( 6 \)[/tex]. Point [tex]\( R \)[/tex] partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a 3:2 ratio. Rachel uses the section formula to find the location of point [tex]\( R \)[/tex] on the number line. Her work is shown below.

Let [tex]\( m = 3 \)[/tex], [tex]\( n = 2 \)[/tex], [tex]\( x_1 = -2 \)[/tex], and [tex]\( x_2 = 6 \)[/tex].

1. [tex]\( R = \frac{m x_2 + n x_1}{m + n} \)[/tex]
2. [tex]\( R = \frac{3(6) + 2(-2)}{3 + 2} \)[/tex]

What is the location of point [tex]\( R \)[/tex] on the number line?

A. [tex]\(\frac{14}{5}\)[/tex]
B. [tex]\(\frac{16}{5}\)[/tex]
C. [tex]\(\frac{18}{5}\)[/tex]
D. [tex]\(\frac{22}{5}\)[/tex]



Answer :

GinnyAnswer
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].

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