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If [tex]\( P, Q \)[/tex], and [tex]\( R \)[/tex] are any three collinear points such that [tex]\( Q \)[/tex] lies between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], find the value of [tex]\( x \)[/tex] if [tex]\( PR = 5x \)[/tex], [tex]\( PQ = 10 \)[/tex], and [tex]\( QR = 3x - 2 \)[/tex].

A. 1
B. 8
C. 6
D. 4

Sagot :

GinnyAnswer
To solve for [tex]\( x \)[/tex] given the collinear points [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] with [tex]\( Q \)[/tex] lying between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], we start with the given distances:

[tex]\[ PR = 5x \][/tex]
[tex]\[ PQ = 10 \][/tex]
[tex]\[ QR = 3x - 2 \][/tex]

Since [tex]\( Q \)[/tex] lies between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], the distance [tex]\( PR \)[/tex] can be expressed as the sum of the distances [tex]\( PQ \)[/tex] and [tex]\( QR \)[/tex]:

[tex]\[ PR = PQ + QR \][/tex]

Substituting the given distances into the equation:

[tex]\[ 5x = 10 + (3x - 2) \][/tex]

First, simplify the right-hand side of the equation:

[tex]\[ 5x = 10 + 3x - 2 \][/tex]
[tex]\[ 5x = 3x + 8 \][/tex]

Next, isolate [tex]\( x \)[/tex] by subtracting [tex]\( 3x \)[/tex] from both sides:

[tex]\[ 5x - 3x = 8 \][/tex]
[tex]\[ 2x = 8 \][/tex]

To find the value of [tex]\( x \)[/tex], divide both sides by 2:

[tex]\[ x = \frac{8}{2} \][/tex]
[tex]\[ x = 4 \][/tex]

Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{4} \)[/tex].
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