How do I work this problem? Flying with the wind, an airplane can travel 1080 miles in six hours, but against the wind, it goes 360 miles in half the time. Find the air speed and wind speed

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To solve this problem, we need to set up a system of equations based on the given information.

Let's assume the speed of the airplane in still air is A miles per hour, and the speed of the wind is W miles per hour.

When the airplane flies with the wind, its effective speed is increased by the speed of the wind. So, the speed becomes (A + W) miles per hour.

Given that the airplane can travel 1080 miles in 6 hours, we can set up the equation:

(A + W) * 6 = 1080

Simplifying this equation, we get:

6A + 6W = 1080

Similarly, when the airplane flies against the wind, its effective speed is decreased by the speed of the wind. So, the speed becomes (A - W) miles per hour.

Given that the airplane can travel 360 miles in half the time, which is 6/2 = 3 hours, we can set up the equation:

(A - W) * 3 = 360

Simplifying this equation, we get:

3A - 3W = 360

Now, we have a system of equations:

6A + 6W = 1080
3A - 3W = 360

To find the values of A (air speed) and W (wind speed), we can solve this system of equations using any suitable method, such as elimination or substitution.

Let's solve the system using the elimination method.

First, we'll multiply the second equation by 2 to make it easier to eliminate the W terms:

6A + 6W = 1080
6A - 6W = 720

Now, we can add the two equations to eliminate the W terms:

(6A + 6W) + (6A - 6W) = 1080 + 720

Simplifying, we get:

12A = 1800

Dividing both sides of the equation by 12, we find:

A = 150

Now, we can substitute this value back into one of the original equations to find W.

Let's substitute it into the first equation:

6(150) + 6W = 1080

Simplifying, we get:

900 + 6W = 1080

Subtracting 900 from both sides, we find:

6W = 180

Dividing both sides by 6, we find:

W = 30

Therefore, the air speed (A) is 150 miles per hour and the wind speed (W) is 30 miles per hour.

plane's normal speed = s

wind speed = w

1080(s+w) = 6
360/(s-w) = 3

6s + 6w = 1080
3s - 3w = 360

12s = 1800
s = 150
w = 30