To solve for the speed of the wind and the speed of the plane in still air, we'll denote the following variables:
- [tex]\( p \)[/tex]: the speed of the plane in still air.
- [tex]\( w \)[/tex]: the speed of the wind.
Given two scenarios:
1. The plane travels 470 miles per hour with the wind, which means [tex]\( p + w = 470 \)[/tex].
2. The plane travels 390 miles per hour against the wind, which means [tex]\( p - w = 390 \)[/tex].
These two equations represent the relationship between the plane’s speed in still air, the wind speed, and the observed speeds with and against the wind:
1. [tex]\( p + w = 470 \)[/tex]
2. [tex]\( p - w = 390 \)[/tex]
To find the values of [tex]\( p \)[/tex] and [tex]\( w \)[/tex], let's solve these equations step-by-step:
### Step 1: Add the Two Equations
We start by adding the two equations together:
[tex]\[ (p + w) + (p - w) = 470 + 390 \][/tex]
Simplifying:
[tex]\[ p + w + p - w = 860 \][/tex]
This reduces to:
[tex]\[ 2p = 860 \][/tex]
Dividing both sides by 2:
[tex]\[ p = 430 \][/tex]
So, the speed of the plane in still air, [tex]\( p \)[/tex], is 430 miles per hour.
### Step 2: Substitute [tex]\( p \)[/tex] into One of the Original Equations
Next, we substitute [tex]\( p \)[/tex] into one of the original equations to solve for [tex]\( w \)[/tex]. We'll use [tex]\( p + w = 470 \)[/tex]:
[tex]\[ 430 + w = 470 \][/tex]
Subtracting 430 from both sides:
[tex]\[ w = 470 - 430 \][/tex]
[tex]\[ w = 40 \][/tex]
So, the speed of the wind, [tex]\( w \)[/tex], is 40 miles per hour.
### Conclusion
The speed of the wind is 40 mph.
