distance, rate, and time

Question

a 555-mile, 5-hour plane trip was flown at two speeds.for the first part of the trip, the average speed was 105 mph.then the tailwind picked up and the remainder of the trip was flown at an average speed of 115 mph.for how long did the plane fly at each speed

Answer 1

So there is two methods to solve the following problem with. The first is Elimination and the second is substitution. I will show you both but the initial starting point are exactly the same.

 

First we need to know that Distance = rate x time because knowing this allows us to set up the system of equations.

 

Now, we know that the trip was 5 hours so, because why not, let's set the 105mph trip time equal to the variable "X" and the 115mph trip time equal to the variable "Y". Which will give us the following equation:

 

X+Y=5hrs

 

and knowing that the distance total is 555 miles that mean the slow mph distance plus fast mph distance must equal 555 giving us this equation:

 

Dx+Dy=555 miles and since Distance = rate x time

 

105mph(X) + 115mph(Y) = 555

 

now we can have our system of equation where one represent the time [1.)] and the other representing the distance [1.)]

 

1.) X+Y = 5

 

2.) 105(X) + 115(Y) = 555

 

Now lets start off with the elimination method. First we get rid of one of the variables by multiplying the simpler equation [1.)] to match with opposite signs. (just watch)

 

1.)-115(X+Y)=5(-115)

2.)105X+115Y=555

 

1.) -115x + -115y = -575

2.) 105X + 115Y = 555

 

Then we smash the two equations together ( add together [1.) + 2.)] )

105X-115X + 115Y-115Y = 555-575

 

-10X = -20

 

Now solve for X

 

-10X (-1/10) = -20 (-1/10)

X= 2 meaning 2 hrs for the slower flight and since it added up to equal 5 this must mean it is 3 hours for the 115mph trip.

 

Now lets do this the substitution method. Which is simply just getting one variable by itself on one equation and plugging it into the other equation.

 

1.) X+Y = 5

2.) 105(X) + 115(Y) = 555

 

From equation 1.) we can get Y by itself by subtracting X to both side

 

1.) Y=5-X

2.)105X+115Y=555

 

Now we plug equation 1.) (since Y=5-X) into equation 2.)

 

2.) 105X + 115(5-X)=555

 

And as you can see all we have is now is just one variable to solve for.

 

105X + 115(5) - 115X = 555

 

Combining like terms and simplifying

 

-10X + 575=555

 

Getting X by itself by subtracting 575 to both side

 

-10X + 575-575=555-575

 

-10X=-20

 

X=2

 

and just like before Y then must equal 3hrs.

 

WAIT!!!!!!!!!! You should never do a problem without a way to make sure it is correct. To do this all you have to is plug the X and Y value back into the two equations and see if they both agree with it.

 

X=2 and Y=3

 

1.) X+Y=5

thus

1.) 2+3=5 --> 5=5 yup. Last I check 5 does equal 5 so this one is good let's check the other equation.

 

2.) 105X + 115Y=555

thus

2.)105(2) + 115(3) = 555 --> 210 + 345 = 555 --> 555=555 yup this works too. Now you can be totally sure you got it right. 

 

 

Answer 2

d = r · t

First segment:

d = 105(5 - t)

Second segment:

d = 115(t)

105(5 - t) + 115(t) = 555

 

Can you solve for t?