Two trains leave the station at the same time, one heading east and the other west. the eastbound train travels 16 miles per hour faster than the westbound train. if the two trains are 800 miles apart after 4 hours, what is the rate of the eastbound train? do not do any rounding.

We'll use variables to represent the speeds of the eastbound and westbound trains.

x will represent the speed of the eastbound train.
y will represent the speed of the westbound train.

The eastbound train is 16 mph faster than the westbound train. An equation can be made from this:



Subtraction is used, because it represents the difference in distances between the two trains if they travel the same direction.

After 4 hours, the trains are 800 miles apart. An equation can be made from this:



Addition is used, because the trains are heading in opposite directions, which means their distances from the starting point are added together.

Set the two equations up vertically:




We will use elimination to solve for x.

Multiply the entire first equation by 4 so that the coefficients for y will be opposite numbers:






Combine the two equations together to cancel out y:



Divide both sides by 8 to get x by itself:



The speed of the eastbound train is 108 mph.

Let's start with what we know:

The area of the rectangle is 54cm² which means: length x width = 54cm²
For the sake of this problem let's refer to length as l and width as w.
We are also trying to find x so let's set up some equations based on the info above.

You said that the length is 2cm more than x. That can be written as this:
l = 2 + x

And you said that the width is 5cm less than twice x. That be written as this:
w = 2x - 5

Let's take those two equations and substitute them into our very first equation (l x w = 54cm²)

l x w = 54cm²
l = 2 + x and w = 2x - 5, so:

(2 + x) x (2x - 5) = 54

Let's use the FOIL method to flesh this out. Remember that FOIL stands for first, outer, inner, last:

First: (2 × 2x)
Outer: (2 × -5)
Inner: (x × 2x)
Last: (x × -5)

That would appear in the equation like this:

(2 × 2x) + (2 × -5) + (x × 2x) + (x × -5) = 54

Time to simplify!

4x - 10 + 2x² - 5x = 54
Which can be simplified further to:

2x² - x - 10 = 54
Subtract 54 from each side:

2x² - x - 10 - 54 = 54 -54
And simplify:

2x² - x - 64 = 0
That's a quadratic equation which means we can solve for x using the quadratic formula which is this:

-b 

Looking at our equation we know that a = 2, b = -1, and c = -64 so let's plug those values in!



Time to simplify:




That's a messy number, but it's what we got! So let's do the + / -

(1 + 22.65)/4 ≈ 5.91
(1 - 22.65)/4 ≈ -5.41

So after the quadratic formula, we believe that x ≈either 5.91 or -5.41
Let's test the two to see which one works.

Remember our earlier equations? Let's use those here.

l = 2 + x, so:
l ≈ 2 + 5.91
l ≈ 7.91cm

And w = 2x - 5, so:
w ≈ 2(5.91) - 5
w ≈ 11.82 - 5
w ≈ 6.82cm

Let's see if that come's close to our area of 54cm²

l x w = 54cm²
7.91cm x 6.82cm = 53.95cm²
If we round that up, it will equal 54cm² so we know that the x-value of 5.91 is the correct answer without even having to test -5.41.

 x ≈ 5.91