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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 -2 and [tex]$S$[/tex] at 6. 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
Given a number line with endpoints [tex]\( Q \)[/tex] at [tex]\(-2\)[/tex] and [tex]\( S \)[/tex] at [tex]\( 6 \)[/tex], we are to find the location of point [tex]\( R \)[/tex] that partitions the directed line segment [tex]\( QS \)[/tex] in a [tex]\( 3:2 \)[/tex] ratio.

Let's use the section formula to determine [tex]\( R \)[/tex].

#### Step-by-Step Solution:

1. Assign the given values:
[tex]\[ Q \implies x_1 = -2 \quad \text{and} \quad S \implies x_2 = 6 \][/tex]
The ratio given is [tex]\( 3:2 \)[/tex].
[tex]\[ \Rightarrow m = 3 \quad \text{and} \quad n = 2 \][/tex]

2. The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( QS \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R = \frac{m x_2 + n x_1}{m+n} \][/tex]

3. Substituting the values into the section formula:
[tex]\[ R = \frac{3 \cdot 6 + 2 \cdot (-2)}{3 + 2} \][/tex]

4. Simplify the expression step-by-step:
[tex]\[ R = \frac{18 + (-4)}{5} \][/tex]
[tex]\[ R = \frac{18 - 4}{5} \][/tex]
[tex]\[ R = \frac{14}{5} \][/tex]

Therefore, the location of point [tex]\( R \)[/tex] on the number line is:
[tex]\[ R = \frac{14}{5} \][/tex]

Among the options given, the correct location of point [tex]\( R \)[/tex] is:
[tex]\[ \boxed{\frac{14}{5}} \][/tex]