Write a system of equations in two variables to solve problem

Write a system of equations in two variables to solve problem.

Flying with a tailwind, a pilot flew an airplane 900 miles in 4.5 hours. Flying into a headwind, the return trip took 5 hours. Find the speed of the plane in calm air and the speed of windÂÂ

Speed in calm air mph?

Speed of wind mph?

The correct answer and explanation is :

To solve this problem, we need to create a system of equations to represent the situation, where:

• Let $x$ be the speed of the plane in calm air (in miles per hour).
• Let $y$ be the speed of the wind (in miles per hour).

Step 1: Define the Relationships

1. Flying with a tailwind:

• The plane’s effective speed with the tailwind is $x + y$, where $y$ is the speed of the wind.
• The plane flew 900 miles in 4.5 hours.
• Using the formula $\text{distance} = \text{speed} \times \text{time}$, we can write: $$
  900 = (x + y) \times 4.5
  $$ Simplifying: $$
  900 = 4.5x + 4.5y \quad \text{(Equation 1)}
  $$

1. Flying into a headwind:

• The plane’s effective speed with the headwind is $x – y$.
• The return trip was also 900 miles, but it took 5 hours.
• Using the formula again, we can write: $$
  900 = (x – y) \times 5
  $$ Simplifying: $$
  900 = 5x – 5y \quad \text{(Equation 2)}
  $$

Step 2: Solve the System of Equations

We have the following system of equations:

1. $900 = 4.5x + 4.5y$
2. $900 = 5x – 5y$

Solve for $x$ and $y$:

1. From Equation 1: $$
   900 = 4.5x + 4.5y \quad \Rightarrow \quad 4.5x + 4.5y = 900 \quad \Rightarrow \quad x + y = 200 \quad \text{(Divide through by 4.5)}
   $$
2. From Equation 2: $$
   900 = 5x – 5y \quad \Rightarrow \quad 5x – 5y = 900 \quad \Rightarrow \quad x – y = 180 \quad \text{(Divide through by 5)}
   $$

Now we have the system:

$$
x + y = 200 \quad \text{(Equation 3)}
$$

$$
x – y = 180 \quad \text{(Equation 4)}
$$

3. Solve by addition:
   Add Equation 3 and Equation 4 to eliminate $y$: $$
   (x + y) + (x – y) = 200 + 180
   $$ $$
   2x = 380 \quad \Rightarrow \quad x = 190
   $$
4. Substitute $x = 190$ into Equation 3: $$
   190 + y = 200 \quad \Rightarrow \quad y = 200 – 190 = 10
   $$

Final Answer:

• The speed of the plane in calm air is $\boxed{190}$ miles per hour.
• The speed of the wind is $\boxed{10}$ miles per hour.

Explanation:

This problem involves using the relationships between speed, time, and distance to form two equations based on the conditions provided. By translating the problem into a system of linear equations, we can use algebraic techniques like substitution or elimination to solve for the unknown variables. The system we set up accounted for the tailwind and headwind effects, and through systematic solving, we determined that the speed of the plane in calm air is 190 miles per hour and the speed of the wind is 10 miles per hour. This approach illustrates how understanding basic equations can help solve real-world problems involving motion.