An Aircraft Travels With The Wind For 120 Miles In 0.75 Of An Hour. The Return Trip Is Flown Against The Wind And Takes Exactly 1 Hour.Which System Of Linear Equations Represents { X $} , T H E S P E E D O F T H E P L A N E I N M I L E S P E R H O U R , A N D \[ , The Speed Of The Plane In Miles Per Hour, And \[ , T H Es P Ee D O F T H E Pl An E Inmi L Es P Er H O U R , An D \[

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Introduction

In this article, we will explore a classic problem in mathematics that involves the speed of an aircraft traveling with and against the wind. We will use this problem to create a system of linear equations and solve for the speed of the plane. This problem is a great example of how mathematics can be used to model real-world situations and solve problems.

The Problem

An aircraft travels with the wind for 120 miles in 0.75 of an hour. The return trip is flown against the wind and takes exactly 1 hour. We are asked to find the speed of the plane, denoted by { x $}$, in miles per hour.

Understanding the Problem

To solve this problem, we need to understand the relationship between the speed of the plane, the speed of the wind, and the time it takes to travel with and against the wind. Let's denote the speed of the wind as { w $}$. When the plane travels with the wind, its effective speed is the sum of its own speed and the speed of the wind, i.e., { x + w $}$. On the other hand, when the plane travels against the wind, its effective speed is the difference between its own speed and the speed of the wind, i.e., { x - w $}$.

Creating the System of Linear Equations

We can now create a system of linear equations based on the information given in the problem. Let's denote the speed of the plane as { x $}$ and the speed of the wind as { w $}$. We know that the plane travels 120 miles in 0.75 of an hour with the wind, so we can write the equation:

1200.75=x+w\frac{120}{0.75} = x + w

Simplifying this equation, we get:

160=x+w160 = x + w

We also know that the plane travels 120 miles in 1 hour against the wind, so we can write the equation:

1201=x−w\frac{120}{1} = x - w

Simplifying this equation, we get:

120=x−w120 = x - w

Solving the System of Linear Equations

We now have a system of two linear equations with two variables:

160=x+w160 = x + w

120=x−w120 = x - w

We can solve this system of equations using the method of substitution or elimination. Let's use the elimination method. If we add the two equations, we get:

280=2x280 = 2x

Dividing both sides by 2, we get:

x=140x = 140

Now that we have found the value of { x $}$, we can substitute it into one of the original equations to find the value of { w $}$. Let's use the first equation:

160=140+w160 = 140 + w

Subtracting 140 from both sides, we get:

20=w20 = w

Conclusion

In this article, we used a classic problem in mathematics to create a system of linear equations and solve for the speed of the plane. We found that the speed of the plane is 140 miles per hour and the speed of the wind is 20 miles per hour. This problem is a great example of how mathematics can be used to model real-world situations and solve problems.

Key Takeaways

  • The speed of the plane is 140 miles per hour.
  • The speed of the wind is 20 miles per hour.
  • When the plane travels with the wind, its effective speed is the sum of its own speed and the speed of the wind.
  • When the plane travels against the wind, its effective speed is the difference between its own speed and the speed of the wind.

Further Reading

If you want to learn more about systems of linear equations and how to solve them, I recommend checking out the following resources:

  • Khan Academy: Systems of Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Systems of Linear Equations

I hope this article has been helpful in understanding how to create and solve systems of linear equations. If you have any questions or comments, please don't hesitate to reach out.

Introduction

In our previous article, we explored a classic problem in mathematics that involves the speed of an aircraft traveling with and against the wind. We created a system of linear equations and solved for the speed of the plane. In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the speed of the plane?

A: The speed of the plane is 140 miles per hour.

Q: What is the speed of the wind?

A: The speed of the wind is 20 miles per hour.

Q: Why do we need to consider the speed of the wind?

A: We need to consider the speed of the wind because it affects the effective speed of the plane when it travels with and against the wind. When the plane travels with the wind, its effective speed is the sum of its own speed and the speed of the wind. On the other hand, when the plane travels against the wind, its effective speed is the difference between its own speed and the speed of the wind.

Q: How do we create a system of linear equations for this problem?

A: We create a system of linear equations by using the information given in the problem. We know that the plane travels 120 miles in 0.75 of an hour with the wind, so we can write the equation:

1200.75=x+w\frac{120}{0.75} = x + w

Simplifying this equation, we get:

160=x+w160 = x + w

We also know that the plane travels 120 miles in 1 hour against the wind, so we can write the equation:

1201=x−w\frac{120}{1} = x - w

Simplifying this equation, we get:

120=x−w120 = x - w

Q: How do we solve a system of linear equations?

A: We can solve a system of linear equations using the method of substitution or elimination. In this case, we used the elimination method. If we add the two equations, we get:

280=2x280 = 2x

Dividing both sides by 2, we get:

x=140x = 140

Q: What is the significance of this problem?

A: This problem is a great example of how mathematics can be used to model real-world situations and solve problems. It also illustrates the importance of considering the speed of the wind when calculating the effective speed of an aircraft.

Q: Can this problem be applied to other real-world situations?

A: Yes, this problem can be applied to other real-world situations where the speed of an object is affected by the speed of another object. For example, it can be used to calculate the speed of a boat traveling with and against the current.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking for extraneous solutions
  • Not using the correct method of solution (e.g. substitution or elimination)
  • Not simplifying the equations before solving
  • Not checking the solution for consistency with the original problem

Conclusion

In this article, we answered some of the most frequently asked questions related to the problem of an aircraft traveling with and against the wind. We hope that this article has been helpful in understanding how to create and solve systems of linear equations. If you have any further questions or comments, please don't hesitate to reach out.

Key Takeaways

  • The speed of the plane is 140 miles per hour.
  • The speed of the wind is 20 miles per hour.
  • When the plane travels with the wind, its effective speed is the sum of its own speed and the speed of the wind.
  • When the plane travels against the wind, its effective speed is the difference between its own speed and the speed of the wind.
  • This problem can be applied to other real-world situations where the speed of an object is affected by the speed of another object.

Further Reading

If you want to learn more about systems of linear equations and how to solve them, I recommend checking out the following resources:

  • Khan Academy: Systems of Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Systems of Linear Equations