Answer:
The distance is [tex]S_1 = 45 \ miles[/tex]
Step-by-step explanation:
From the question we are told that
The speed at which Gabby rode his bike is [tex]v = 18 \ miles / hour[/tex]
The average speed of the car is [tex]u = 60 \ miles / hours[/tex]
The distance from Gabby's house to the parents house is [tex]d = 435 \ miles[/tex]
The time taken for Gabby to get to his parents house is t = 9 hours
Generally the distance from Gabby's house to his parent house is mathematically reparented as
[tex]S_1 + S_2 = d[/tex]
Here [tex]S_1[/tex] is the distance from Gabby's house to his sisters house which is mathematically represented as
[tex]S_1 = t_1 * v[/tex]
=> Here [tex]t_1[/tex] is the time taken to move from Gabby's house to his sisters house
while [tex]S_2[/tex] is the distance from his sisters house to his parents house which is mathematically represented as
[tex]S_2 = t_2 * u[/tex]
So
[tex]vt_1 + ut_2 = d[/tex]
=> [tex]18t_1 + 60t_2 = 435 --- (1)[/tex]
Generally the total time taken to arrive his parents house from his house is mathematically represented as
[tex]t_1 + t_2 = t[/tex]
=> [tex]t_1 + t_2 = 9 --- (2)[/tex]
Solving equation 1 and 2 simultaneously we have that
[tex]18t_1 + 60t_2 = 435 --- (1)[/tex]
[tex]t_1 + t_2 = 9 --- (2)[/tex]
multiply equation 2 by 18 and equation 1 by 1
[tex]18t_1 + 60t_2 = 435 --- (3)[/tex]
[tex]18t_1 +18 t_2 = 162 --- (4)[/tex]
Substracting equation 3 from 4
[tex]0t_1 + 42t_2 = 273[/tex]
=> [tex]t_2 = \frac{273}{42}[/tex]
=> [tex]t_2 = 6.5 \ hours[/tex]
substituting [tex]t_2[/tex] into equation 2
[tex]t_1 + 6.5 = 9 --- (2)[/tex]
=> [tex]t_1 = 2.5 \ hours[/tex]
So the distance from Gabby's house to his sisters house is mathematically represented as
[tex]S_1 =2.5 * 18[/tex]
=> [tex]S_1 = 45 \ miles[/tex]
