Elena went on a 6-mile walk. She completed the first half of the walk 1 mi/h faster than usual and the second half of the walk 2 mi/h slower than the first half.

a. If it took her 7 .2 h to complete the walk, what is her usual rate?
b. What is the formula necessary to solve this problem?

if her usual speed is x, then since time = distance/speed,

3/(x+1) + 3/(x-1) = 7.2

To solve this problem, we can use the formula:

Distance = Rate × Time

Let's break down the information given step by step:

a. Elena completed the first half of the walk 1 mi/h faster than her usual rate. Let's assume her usual rate is 'x' mi/h. Therefore, her rate for the first half of the walk is (x + 1) mi/h.

b. The second half of the walk was completed 2 mi/h slower than the first half. So, her rate for the second half of the walk is ((x + 1) - 2) mi/h, which simplifies to (x - 1) mi/h.

c. We know that Elena completed the entire 6-mile walk in 7.2 hours.

Now, let's use the formula Distance = Rate × Time for each half of the walk:

For the first half:
Distance = (x + 1) mi/h * t1 (where t1 is the time taken to complete the first half)

For the second half:
Distance = (x - 1) mi/h * t2 (where t2 is the time taken to complete the second half)

Since the entire walk is 6 miles, we have:
(x + 1) mi/h * t1 + (x - 1) mi/h * t2 = 6

We also know that the total time taken to complete the walk is 7.2 hours, so:
t1 + t2 = 7.2

Now we have a system of two equations:
(x + 1) * t1 + (x - 1) * t2 = 6
t1 + t2 = 7.2

Solving this system of equations will give us the usual rate 'x'.

To solve this system, we have a few options: substitution, elimination, or graphing. One common method is substitution, which involves solving one equation for one variable and substituting it into the other equation.