Problem 27 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A small single-engine plane travels 150 miles per hour with a tailwind and 90 miles per hour with a headwind. Find the speed of the wind and the speed of the plane in still air.

Step-by-Step Solution

Answer

The speed of the plane in still air is 120 mph, and the speed of the wind is 30 mph.

Step 1 - Define variables

Let the speed of the plane in still air be denoted as \( p \) (in miles per hour), and let the speed of the wind be denoted as \( w \) (in miles per hour).

Step 2 - Write equations based on the given information

With the tailwind, the plane's speed is \( p + w = 150 \) miles per hour. With the headwind, the plane's speed is \( p - w = 90 \) miles per hour.

Step 3 - Form a system of equations

The two equations are: 1. \( p + w = 150 \)2. \( p - w = 90 \)

Step 4 - Add the equations to eliminate \( w \)

Add the two equations: \( (p + w) + (p - w) = 150 + 90 \)This simplifies to: \( 2p = 240 \)

Step 5 - Solve for \( p \)

Divide both sides by 2: \( p = \frac{240}{2} = 120 \)So, the speed of the plane in still air is 120 miles per hour.

Step 6 - Substitute \( p \) back into one of the equations

Using the equation \( p + w = 150 \), substitute \( p \) with 120:\( 120 + w = 150 \)Solve for \( w \): \( w = 150 - 120 = 30 \)So, the speed of the wind is 30 miles per hour.

Key Concepts

Defining Variables

Understanding word problems in algebra starts with defining variables. Variables are symbols used to represent unknown values.
In this exercise, we are asked to find the speed of the plane in still air and the speed of the wind.
Step 1: Define the unknowns
Let's use the letter p to represent the speed of the plane in still air (in miles per hour).
We'll use the letter w to represent the speed of the wind (in miles per hour).

By defining variables, we turn a word problem into a mathematical one. This sets the foundation for forming equations.

System of Equations

Once variables are defined, the next step is to establish a system of equations. This system represents the given conditions of the problem.
Step 2: Write the equations
With a tailwind, the speed of the plane increases due to the wind, forming the equation:
\( p + w = 150 \)
With a headwind, the speed of the plane decreases due to the wind, forming the equation:
\( p - w = 90 \)

Step 3: Form the system of equations
Combining these equations, we have a system:
\( p + w = 150 \)
\( p - w = 90 \)
A system of equations consists of two or more equations with the same variables. These equations work together to help us find the values of the unknowns.

Solving Linear Equations

To find the values of the variables, we solve the system of linear equations.
Step 4: Add the equations
By adding the two equations, we eliminate one variable:
\( (p + w) + (p - w) = 150 + 90 \)
This simplifies to
\( 2p = 240 \)

Step 5: Solve for p
Divide by 2:
\( p = \frac{240}{2} = 120 \)
So, the speed of the plane in still air is 120 miles per hour.

Step 6: Substitute p and solve for w
Replace p in the first equation:
\( 120 + w = 150 \)
\( w = 150 - 120 = 30 \)
So, the speed of the wind is 30 miles per hour.

Breaking down each step makes solving linear equations straightforward. We start by eliminating one variable, then solve for the other. Finally, we substitute back to find the remaining unknown.