# System of Equations: Solving Rates of Planes 600 Miles Apart

• MHB
• mathgeek7365
In summary: Solve them for r, and then calculate r+5.In summary, we have two planes flying from one city to another, with a distance of 600 miles. The slower plane flies at a rate of r mph, while the faster plane flies at a rate of r+5 mph. The slower plane takes 2 hours longer to reach the destination. Using the formula d=rt, we can create a system of equations and solve for the rate of each plane.
mathgeek7365
"Two planes leave a city for another city that us 600 miles away. One of the planes is flying 50 miles per hour faster than the other. The slower plane takes 2 hours longer to reach the city. What is the rate of each plane? Write and solve a system of equations."

My daughter is well aware that d=rt, where d represents distance, r represents rate, and t represents time. She also knows how to solve systems of equations. She is unsure on how to create the system of equations from the information given. She would like some hints as to how to start/she would like some help getting on the right track. Thanks.

Hello, mathgeek7365!

"Two planes leave a city for another city that us 600 miles away.
One of the planes is flying 50 miles per hour faster than the other.
The slower plane takes 2 hours longer to reach the city. What is
the rate of each plane? Write and solve a system of equations."

My daughter is well aware that d = rt, where d represents distance,
r represents rate, and t represents time. She also knows how to solve
systems of equations. She is unsure on how to create the system of
equations from the information given.

We will use: $$\; d\,=\,rt \quad\Rightarrow\quad t \,=\,\frac{d}{r}$$

The slower plane flies at $$r$$ mph.
The faster plane flies at $$r\!+\!5$$ mph.

The faster plane flies 600 miles at $$r\!+\!5$$ mph.
This takes:$$\:\tfrac{600}{r+5} \:=\:t$$ hours.

The slower plane flies 600 miles at $$r$$ mph.
This takes: $$\:\tfrac{600}{r} \:=\:t+2$$ hours.

There are the two equations.

## 1. How do you solve a system of equations for finding the rates of two planes 600 miles apart?

To solve this problem, you can use the distance formula, which is distance = rate x time. Since we know the distance between the two planes is 600 miles, we can set up two equations: 600 = rate of plane A x time and 600 = rate of plane B x time. Then, we can use either substitution or elimination method to solve for the rates of both planes.

## 2. What is the distance formula and how is it used in solving this problem?

The distance formula is distance = rate x time. In this problem, we can use it to set up two equations based on the given information: 600 = rate of plane A x time and 600 = rate of plane B x time. By plugging in the distance (600 miles) and solving for the rates, we can find the speeds of both planes.

## 3. Can you use the Pythagorean theorem to solve this problem?

No, the Pythagorean theorem is used to find the distance between two points on a coordinate plane. Since this problem involves two moving objects, we need to use the distance formula and set up equations to find the rates of the planes.

## 4. What other factors may affect the rates of the two planes?

There are a few factors that may affect the rates of the two planes, such as wind speed and direction, air traffic control, and speed restrictions. These factors may also change during the flight, so the rates calculated using the distance formula may not reflect the actual speeds of the planes.

## 5. Are there any real-life applications of this problem?

Yes, this type of problem can be applied to air traffic control and flight planning. By calculating the rates of two planes, air traffic controllers can ensure that the planes maintain a safe distance from each other and plan flight paths accordingly. It can also be used in calculating the fuel consumption and flight time of airplanes.

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