Problem 286 - Elementary Algebra

Question

In the following exercises, solve. A commercial jet and a private airplane fly from Denver to Phoenix. It takes the commercial jet 1.1 hours for the flight, and it takes the private airplane 1.8 hours. The speed of the commercial jet is 210 miles per hour faster than the speed of the private airplane. Find the speed of both airplanes.

Step-by-Step Solution

Answer

Speed of the private airplane is 330 mph. Speed of the commercial jet is 540 mph.

Step 1: Define Variables

Let the speed of the private airplane be denoted as \( v \text{ mph} \). Therefore, the speed of the commercial jet will be \( v + 210 \text{ mph} \).

Step 2: Set Up Distance Formulas

The distance covered is the same for both planes. Using the formula \( \text{distance} = \text{speed} \times \text{time} \), we can write the distances as: \[ \text{Distance for private airplane} = v \times 1.8 \text{ Distance for commercial jet} = (v + 210) \times 1.1 \]

Step 3: Set Up Equation

Since the distances are the same, set the expressions equal to each other: \[ v \times 1.8 = (v + 210) \times 1.1 \]

Step 4: Solve for v

Expand and solve for \( v \): \[ 1.8v = 1.1v + 231 \] \[ 1.8v - 1.1v = 231 \] \[ 0.7v = 231 \] \[ v = \frac{231}{0.7} \] \[ v = 330 \]

Step 5: Find the Speed of the Commercial Jet

Use the value of \( v \) to find the speed of the commercial jet: \[ \text{Speed of commercial jet} = v + 210 = 330 + 210 = 540 \text{ mph} \]

Key Concepts

solving linear equations

Linear equations are equations where the variable is raised to the power of one. These equations have a constant rate of change and graph as a straight line.
In our problem, we defined the speed of the private airplane as v. We set up the equation by using the distance-speed-time relationship and equating the distances for both the private airplane and commercial jet. Here’s how to solve the equation step-by-step:

• First, we write the equation: \[ 1.8v = (v + 210) \times 1.1 \]
• Next, we expand the equation: \[ 1.8v = 1.1v + 231 \]
• Then, subtract 1.1v from both sides to combine like terms: \[ 1.8v - 1.1v = 231 \]
• The resulting equation is: \[ 0.7v = 231 \]
• Solve for v by dividing by 0.7: \[ v = \frac{231}{0.7} = 330 \]

Hence, the speed of the private airplane is 330 mph. Solving linear equations involves steps like isolating the variable, combining like terms, and using operations to simplify the equation. Remember to check your work to ensure the solution is correct.

distance-speed-time relationship

One fundamental concept in algebra and physics is the relationship between distance, speed, and time. The formula to remember is:
\[ \text{distance} = \text{speed} \times \text{time} \] This equation helps us understand how far an object travels over a period of time at a given speed.
In our exercise, we used this relationship to find the distances traveled by both the commercial jet and the private airplane.

• For the private airplane: \[ \text{Distance} = v \times 1.8 \]
• For the commercial jet: \[ \text{Distance} = (v + 210) \times 1.1 \]

Since both airplanes travel the same distance from Denver to Phoenix, we set the two expressions equal to each other. Doing this allows us to solve for the speed of each plane. Be sure to use consistent units and double-check calculations to avoid mistakes.

variable definition in algebra

Defining variables is a critical first step in algebraic problem solving. A variable is a symbol, often a letter, that represents an unknown value. Proper variable definition sets the stage for forming equations and finding solutions.
In our problem, we defined the speed of the private airplane as v. This variable helps us express relationships between different quantities.
When defining variables:

• Be clear and specific. For example, let v = the speed of the private airplane in mph.
• Use the context of the problem to guide your definitions.
• Keep track of units to ensure consistency.

Properly defined variables make setting up and solving equations much more straightforward. They provide clarity and prevent confusion as you work through the problem. In this case, we used v to find speeds that satisfy both the commercial jet and the private airplane.