Given that:
A is the midpoint of line segment PR.
PA = 7x - 11 and PR = 8x +14.
Solution:
Since A is the midpoint of line segment PR.
[tex]PA=AR=7x-11[/tex]
Using segment addition property, we get
[tex]PR=PA+AR[/tex]
[tex]8x+14=(7x-11)+(7x-11)[/tex]
[tex]8x+14=14x-22[/tex]
Isolate variable terms.
[tex]14+22=14x-8x[/tex]
[tex]36=6x[/tex]
Divide both sides by 6.
[tex]\dfrac{36}{6}=x[/tex]
[tex]6=x[/tex]
The value of x is 6.
Now,
[tex]AR=7x-11[/tex]
[tex]AR=7(6)-11[/tex]
[tex]AR=42-11[/tex]
[tex]AR=31[/tex]
Therefore, the value of line segment AR is 31 units.
