Steve hikes from the campground to the waterfall at a steady pace after two hours he’s 13 miles from the waterfall and after four hours he’s 6 miles from the waterfall

We know that Steve hikes at a steady pace, from this we can assume that he hikes at a constant velocity, then we can model this situation with a linear relationship:

A linear relationship can be written as:

y = a*x + b

where a is the slope and b is the y-axis intercept.

For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:

a = (y2 - y1)/(x2 - x1).

where y represents the position and x the time.

Then first let's define our system.

Let's define the waterfall as the 0mi in the position axis.

Let's define the moment when he starts hiking (from the campground) as the 0h in the time axis.

now we know that:

"After two hours he’s 13 miles from the waterfall"

This can be modeled with the point (2h, 13mi)

"After four hours he’s 6 miles from the waterfall"

This can be modeled with the point (4h, 6mi)

Now we have the two points we wanted, with this we can find the slope, that in this case, represents the hiking speed of Steve:

a = (6mi - 13mi)/(4h - 2h) = -3.5mi/h

We have a negative quantity, this is because we defined the waterfall as the zero in the position axis, so he is moving towards the zero, that's why we have a negative slope.

Then the position can be written as:

y = (-3.5mi/h)*x + b

Now to find the value of b, we can replace one of our points in the equation, for example with the first point we have:

x = 2h, y = 13mi

13mi = (-3.5mi/h)*2h + b

13mi = -7mi + b

13mi + 7mi = b

20mi = b.

Then the equation is:

y = (-3.5mi/h)*t + 20mi

From this, we know that the campground (the initial position of Steve) is 20 miles away from the waterfall.

And with this, we also can find the time that Steve needs to get to the waterfall.

Remember that the waterfall was the zero in the position, then when we have:

y = 0 = (-3.5mi/h)*t + 20mi

   -20mi/(-3.5mi/h) = t

     5.7h = t.

This means that Steve will get to the waterfall 5.7 hours after he lefts the campground.