Matt and Anna are frequent fliers on Fast n Go Airplanes. They often fly between two cities that are a distance of 1680 miles apart. On one particular trip, they flew into the wind and the trip took 4 hours. the return trip with the wind behind them, only took about 3 hours. Find the speed of the wind and the speed of the plane in still air .
The Wind Speed is ____ mph
The Speed of the Plane is ____ mph
Also,
-3x - y = 4
y - 4z = 10
x - 4y + z = -12
What is (X, Y, Z)
To find the speed of the wind and the speed of the plane in still air, we can set up a system of equations.
Let's assume the speed of the plane in still air is represented by "p" and the speed of the wind is represented by "w".
1. When flying into the wind:
The effective speed of the plane would be reduced by the speed of the wind, so we have:
p - w = distance/time
p - w = 1680/4
p - w = 420 (equation 1)
2. When flying with the wind:
The effective speed of the plane would be increased by the speed of the wind, so we have:
p + w = distance/time
p + w = 1680/3
p + w = 560 (equation 2)
Now, we can solve this system of equations using any method such as substitution or elimination:
Let's use the elimination method by adding equation 1 and equation 2:
(p - w) + (p + w) = 420 + 560
2p = 980
p = 490
Now we can substitute the value of "p" back into, say, equation 1 to find the value of "w":
490 - w = 420
-w = 420 - 490
-w = -70
w = 70
Therefore, the wind speed is 70 mph, and the speed of the plane in still air is 490 mph.
Now, onto the second problem:
To find the values of x, y, and z, we can solve the system of equations:
-3x - y = 4 (equation 1)
y - 4z = 10 (equation 2)
x - 4y + z = -12 (equation 3)
There are several methods to solve this system of equations, such as substitution, elimination, or matrix methods.
Let's use the substitution method:
From equation 2, we can isolate y:
y = 4z + 10 (equation 4)
Now substitute equation 4 into equation 1:
-3x - (4z + 10) = 4
-3x - 4z - 10 = 4
-3x - 4z = 4 + 10
-3x - 4z = 14 (equation 5)
Similarly, substitute equation 4 into equation 3:
x - 4(4z + 10) + z = -12
x - 16z - 40 + z = -12
x - 15z - 40 = -12
x - 15z = -12 + 40
x - 15z = 28 (equation 6)
Now we have two equations with two unknowns (x and z). We can solve them simultaneously.
Next, let's multiply equation 5 by 3:
-9x - 12z = 42 (equation 7)
Now, add equation 6 and equation 7:
(x - 15z) + (-9x - 12z) = 28 + 42
x - 9x - 15z - 12z = 70
-8x - 27z = 70 (equation 8)
Now, we have two equations with two unknowns (x and z). We can solve them simultaneously.
Let's multiply equation 8 by -3:
24x + 81z = -210 (equation 9)
Now, add equation 8 and equation 9:
24x + 81z + (-8x - 27z) = -210 + 70
16x + 54z = -140 (equation 10)
Now, we have two equations with two unknowns (x and z). We can solve them simultaneously.
Let's multiply equation 10 by -2:
-32x - 108z = 280 (equation 11)
Now, add equation 9 and equation 11:
-32x - 108z + (24x + 81z) = 280 + (-210)
-8x - 27z = 70
-8x - 27z = -8 (equation 12)
Now, equation 12 shows that -8x - 27z = -8. We can solve this equation to find the values of x and z:
-8x - 27z = -8
Let's assume a value for x (say, 1):
-8(1) - 27z = -8
-8 - 27z = -8
-27z = 0
z = 0
Now, substitute the value of z back into equation 8:
-8x - 27(0) = 70
-8x = 70
x = -8.75
Finally, substitute the values of x and z back into equation 4 to find the value of y:
y = 4z + 10
y = 4(0) + 10
y = 10
Therefore, the values of (x, y, z) are (-8.75, 10, 0).