The speed of a plane in still air is x miles per hour. The wind blows at a constant speed of y miles per hour in an unvarying direction. Flying into a headwind (against the wind), the plane takes 4 hours to travel 500 miles. Flying with a tailwind (with the wind), the plane takes 3 hours to travel 465 miles. Write a system of linear equations to represent the scenario.(1 point)

Question

Responses

{4x+y=5003x−y=465
enlarged left brace start layout first row 4 x plus y equals 500 second row 3 x minus y equals 465 end layout

{4x−y=5003x+y=465
enlarged left brace start layout first row 4 x minus y equals 500 second row 3 x plus y equals 465 end layout

{4(x−y)=5003(x+y)=465
enlarged left brace start layout first row 4 left parenthesis x minus y right parenthesis equals 500 second row 3 left parenthesis x plus y right parenthesis equals 465 end layout

{4(x+y)=5003(x−y)=465

Answer 1

To write a system of linear equations representing the scenario, we start by defining the effective speeds of the plane when flying against and with the wind.

1. Flying against the wind:

   • The speed of the plane against the wind is \( x - y \).
   • The distance flown is 500 miles.
   • The time taken is 4 hours.
   • Using the formula: distance = speed × time, we can set up the equation: \[ 4(x - y) = 500 \]

2. Flying with the wind:

   • The speed of the plane with the wind is \( x + y \).
   • The distance flown is 465 miles.
   • The time taken is 3 hours.
   • Again, using the distance = speed × time formula, we get: \[ 3(x + y) = 465 \]

Now we can rewrite these equations:

1. From the first equation: \[ 4x - 4y = 500 \]

2. From the second equation: \[ 3x + 3y = 465 \]

Thus, the system of linear equations is: \[ \begin{align*} 4x - 4y &= 500 \ 3x + 3y &= 465 \end{align*} \]

Now, let's look for the corresponding option from your responses:

• First option: \( {4x + y = 500, 3x - y = 465} \)
• Second option: \( {4x - y = 500, 3x + y = 465} \)
• Third option: \( {4(x - y) = 500, 3(x + y) = 465} \)
• Fourth option: \( {4(x + y) = 500, 3(x - y) = 465} \)

The equations \( 4x - 4y = 500 \) and \( 3x + 3y = 465 \) simplifies to \( 4(x - y) = 500 \) and \( 3(x + y) = 465\), which is option 3.

So the correct answer is: \[ {4(x - y) = 500, 3(x + y) = 465} \]