"psa" means plane speed in still air; "ws" means wind speed
psa -ws = 550
psa +ws = 600
Adding both equations we get
2*psa = 1150
Plane speed in still air = 575
Since psa +ws = 600
By substitution we get wind speed = 25

B) Solving for airplane velocity and distance given wind velocity and time.

Example:
An airplane flying with a 40 mph wind takes 4 hours to make a trip.
On the return trip, the airplane flies against a 40 mph wind and takes 4.5 hours to make the trip.
What is the airplane velocity and the distance travelled (one way)?

Using the calculator, we click "B" then enter
Wind Velocity   40
Time 1   4
Time 2   4.5
(entering Time 1 = 4.5   Time 2 = 4 will also work)

Clicking "Calculate" we see the answers are:
Airplane Velocity   680
Distance   2880

Without using the calculator:
Distance = velocity * time
Distance = (plane velocity + wind velocity) * time
Dist (out) = (plane vel + 40) * 4
Dist (return) = (plane vel - 40) * 4.5

Since this is a round trip, the distance is the same so:
(plane vel + 40) * 4 = (plane vel - 40) * 4.5
4*plane vel + 160 = 4.5 * plane vel - 180
340 = .5 * plane velocity
plane velocity = 680

Putting this amount into this equation:
Dist (out) = (plane vel + 40) * 4
Distance = (680 +40) * 4
Distance = 720 * 4
Distance = 2,880

C) Solving for Distance and Wind Velocity given airplane velocity and time.

Example: You take a round-trip on an airplane, that has a velocity in still air of 440 mph.
On your trip out, the plane flies for 6 hours against the wind and on your return trip, it flies for 5 hours with the wind.
What is the wind speed and the one-way distance it traveled?

Using the calculator, we click "C" then enter

Airplane Velocity   440
Time 1   5
Time 2   6
(entering Time 1 = 6   Time 2 = 5 will also work)

Clicking "Calculate" we see the answers are:
Distance   2400
Wind Velocity   40

Without using the calculator,
Distance = velocity * time
Distance = (plane velocity + wind velocity) * time
Distance (out) = (440 - wind velocity) * 6
Distance (return) = (440 + wind velocity) * 5

Since it's a round trip, the distance is the same so:
(440 - wind velocity) * 6 = (440 + wind velocity) * 5
-6* wv + 2,640 = 2,200 +5*wv
440 = 11*wv
wind velocity = 40

Putting this value into this equation:
Distance (out) = (440 - wind velocity) * 6
Distance = (440 - 40) * 6
Distance = (400) * 6
Distance = 2,400

D) Solving for Wind Velocity given Airplane Velocity and Distance.

Example: A plane's velocity in still air is 210 miles per hour.
It flies for 725 miles with the wind and in the same amount of time, it flies 675 miles against the wind.
What is the wind velocity?

Using the calculator, we click "D" then enter

Airplane Velocity   210
Distance 1   725
Distance 2   675
Entering Distance 1 = 675 and Distance 2 = 725 will also work

Clicking "Calculate" we see the answer is:
Wind Velocity   7.5

Without using the calculator
There are 2 ways to do this.
If you want the complete explanation (the solution that algebra teachers like), then just keep reading.
Otherwise scroll to "the shorter method".
We are not given a specific amount of time, but we do know that time = distance ÷ rate
Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other.

Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity)

725 ÷ (210 + wind velocity) = 675 ÷ (210 - wind velocity)
(725 ÷ 675) = (210 + wind velocity) ÷ (210 - wind velocity)
(725 × 210) - (725 × wind velocity) = (675 × 210) + (675 × wind velocity)
(725 × 210) - (675 × 210) = (725 × wind velocity) + (675 × wind velocity)
152,250 - 141,750 = 1,400 wind velocity
10,500 = 1,400 wind velocity
wind velocity = 7.5 miles per hour

The shorter method:
When using this method, make sure distance1 is greater than distance2.
We'll also abbreviate plane velocity and wind velocity as pv and wv.
Insert the numbers from the problem into this equation:

[(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity
(152,250 -141,750) ÷ 1,400 = wind velocity
10,500 ÷ 1,400 = wind velocity
wind velocity = 7.5 miles per hour

E) Solving for Plane Velocity given Wind Velocity and Distance.

Example: The wind velocity is 12 miles per hour.
A plane flies for 850 miles with the wind and in the same amount of time, it flies 750 miles against the wind.
What is the airplane velocity?

Using the calculator, we click "E" then enter

Wind Velocity   12
Distance 1   850
Distance 2   750
Entering Distance 1 = 750 and Distance 2 = 850 will also work

Clicking "Calculate" we see the answer is:
Airplane Velocity   192

Without using the calculator
There are 2 ways to do this.
If you want the complete explanation (the solution that algebra teachers like), then just keep reading.
Otherwise scroll to "the shorter method".
We are not given a specific amount of time, but we do know that time = distance ÷ rate
Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other.

Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity)

850 ÷ (plane velocity + 12) = 750 ÷ (plane velocity - 12)
(850 ÷ 750) = [(plane velocity + 12) ÷ (plane velocity - 12)]
(850 × -12) + 850 pv = (750 × 12) + 750 pv
-10,200 + 850 pv = 9,000 + 750 pv
100 pv = 19,200
plane velocity = 192

The shorter method:
When using this method, make sure distance1 is greater than distance2.
We'll also abbreviate plane velocity and wind velocity as pv and wv.
Insert the numbers from the problem into this equation:

[(850 × 12) + (750 × 12)] ÷ (850 -750) = plane velocity
(10,200 + 9,000) ÷ 100 = plane velocity
plane velocity = 192

F) Solving for Boat Velocity when given Current Velocity, Distance and Time.

Example: Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip.
The velocity of the current is 5 miles per hour.
What is the boat velocity?

Using the calculator, we click "F" then enter

Time   8
Distance   30
Wind (or Current) Velocity   5

Clicking "Calculate" we see the answer is:
Airplane (or Boat) Velocity   10

Without using the calculator,
We know that time = distance ÷ velocity.
We only know the total time, so any equations we write must take that into account.

Going upstream, the time would be:
time1 = 30 ÷ (bv - 5)

Going downstream, the time would be:
time2 = 30 ÷ (bv + 5)

One thing we do know is that the total time equals 8 hours.
Since time1 + time2 = 8 hours, then we can say:

[ 30 ÷ (bv - 5) ] + [ 30 ÷ (bv + 5) ] = 8
Multiplying both sides by (bv -5)
30 + [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -40
[ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -70
Multiplying both sides by (bv +5)
30bv -150 = 8bv² -70bv +40bv -350
Collecting the terms
8bv² -60bv -200 = 0
Using the quadratic equation calculator
boat velocity = 10 miles per hour