We have that Liam runs 4 miles per hour and gets a 0.25 mile head start. We can write the relation function of this event like this:
[tex]y=4x+0.25[/tex]
Now, Harper runs 0.5 miles per hour faster than Liam. We can represent this in the following way:
[tex]\begin{gathered} \text{Harper's speed = Liam's speed + 0.5m/h}=4+0.5=4.5m/h \\ \end{gathered}[/tex]
then, the relation function for Harper is:
[tex]y=4.5x[/tex]
Since Liam and Harper run the same distance, we can match both equations to get the following expression:
[tex]4x+0.25=4.5x[/tex]
to find how many hours do they run, we have to solve for x:
[tex]\begin{gathered} 4x+0.25=4.5x \\ \Rightarrow0.25=4.5x-4x \\ \Rightarrow0.25=0.5x \\ \Rightarrow x=\frac{0.25}{0.5}=0.5 \\ x=0.5 \end{gathered}[/tex]
we have that x = 0.5, this means that they both run for half and hour the same distance.
