SOLUTION: A bus with 12 soccer players broke down in a town 20 miles from its destination. The coach’s car was available but could carry only 4 players at a time. Also, it could travel only

We make this diagram, where the broken-down bus is at A and the
destination is at F.  B,C,D, and E are points along the way
with unknown distances v,w,x,y,z miles between them. 

|  v miles  |  w miles  | x miles  |  y miles  |  z miles  |      
------------------------------------------------------------
A           B           C          D           E           F

We of course have the first equation as

             v + w + x + y + z = 20

Here are the 5 trips the car makes.

1. Car drives from A to D. (1st 4 get out and walk from D to F.)
2. Car drives from D to B. (picks up 2nd 4, who have walked from A to B.) 
3. Car drives from B to E. (2nd 4 get out and walk from E to F.) 
4. Car drives from E to C. (picks up 3rd 4, who have walked from A to C.) 
5. Car drives from C to F.  

Now we'll fill in the details:

1. Car takes 1st 4 from A to D, a distance of v+w+x miles.
   When car gets to D, it has traveled v+w+x miles and therefore
   (v+w+x)/20 hours have passed, using TIME = DISTANCE/RATE. 

   [Then the 1st 4 have y+z miles yet to walk to F, which will take them
    (y+z)/4 hours.  So total time = (v+w+x)/20 + (y+z)/4 hours, or
    simplifying, total time = (v+w+x+5y+5z)/20 hours.]  

2. Car drives from D to B, a distance of x+w miles, which takes 
   (x+w)/20 hours more. When car gets to B, (v+w+x)/20 + (x+w)/20,
   or (v+2w+2x)/20 hours have passed. During this time the 2nd and
   3rd 4 have walked from A to B and using Distance=Rate·Time, this 
   is 4(v+2w+2x)/20 or (v+2w+2x)/5 miles.  Since A to B is also v, 
   we have the equation v = (v+2w+2x)/5 which when simplified becomes
   the equation 5v = v+2w+2x or 4v = 2w+2x or 2v = w+x 

3. Car drives from B to E, a distance of w+x+y miles, which takes
   (w+x+y)/20 miles.  When car gets to E, (v+2w+2x)/20 + (w+x+y)/20,
   or (v+3w+3x+y)/20 hours have passed. 

   [Then the 2nd 4 have z miles yet to walk from E to F, which will 
    take them z/4 hours.  So total time = (v+3w+3x+y)/20 + z/4 hours,
    or simplifying, total time = (v+3w+3x+y+5z)/20 hours.] 

    Now we have two expressions for the total time, so we can
    set them equal
    (v+w+x+5y+5z)/20 = (v+3w+3x+y+5z)/20
         v+w+x+5y+5z = v+3w+3x+y+5z
                  4y = 2w+2x
                  2y = w+x
    
4. Car drives from E to C, a distance of y+x miles, which takes 
   (y+x)/20 hours more. When car gets to B, (v+3w+3x+y)/20 + (y+x)/20,
   or (v+3w+4x+2y)/20 hours have passed. During this time the 3rd 4 
   have walked from A to C and using DISTANCE=RATExTIME, this 
   is 4(v+3w+4x+2y)/20 or (v+3w+4x+2y)/5 miles.  Since A to C is also v+w, 
   we have the equation v+w = (v+3w+4x+2y)/5 which when simplified becomes
   the equation 5v+5w = v+3w+4x+2y or 4v+2w=4x+2y or 2v+w = 2x+y.

5. Car drives from C to F, the destination, a distance of x+y+z miles, 
   which takes (x+y+z)/20 hours more. So the total time that has passed
   is (v+3w+4x+2y)/20 + (x+y+z)/20 or (v+3w+5x+3y+z)/20.  So this is a
   third expression for the total time, and we set it equal to one of the
   above expressions for the total time, and get

   (v+3w+5x+3y+z)/20 = (v+w+x+5y+5z)/20
        v+3w+5x+3y+z = v+w+x+5y+5z   
               2w+4x = 2y+4z
                 w+x = y+z

So the equations we have are

v+w+x+y+z=20    
2v = w+x
2y = w+x
2v+w = 2x+y 
w+x = y+z

To solve it by matrices, we write it:

 v + w +  x +  y + z = 20
2v - w -  x          =  0
   - w -  x + 2y     =  0
2v + w - 2x -  y     =  0
     w +  x -  y - z =  0

So we get v=4, w=4, x=4, y=4, z=4

So    

|  4 miles  |  4 miles  | 4 miles  |  4 miles  |  4 miles  |      
------------------------------------------------------------
A           B           C          D           E           F

1. Car drives from A to D. That's 12 miles. 
2. Car drives from D to B. That's 8 miles.  
3. Car drives from B to E. That's 12 miles 
4. Car drives from E to C. That's 8 miles. 
5. Car drives from C to F. That's 12 miles.

So the car went a total of 12+8+12+8+12 or 52 miles.
At 20 miles per hour, that took the car 52/20 or 13/5 hours or 2 3/5 hours

The 1st 4:
Rode 12 miles from A to D, which took them 12/20 or 3/5 hours,
Then walked 8 miles from D to F, which took them 8/4 or 2 hours.
So it took the 1st 4 also 2 3/5 hours to get from A to F     
    
The 2nd 4:
Walked 4 miles from A to B, which took them 4/4 or 1 hour
Rode 12 miles from B to E, which took them 12/20 or 3/5 hour.
Walked 4 miles from E to F, which took them 4/4 or 1 hour.
So it took the 2nd 4 also 2 3/5 hours to get from A to F

The 3rd 4:
Walked 8 miles from A to C, which took them 8/4 or 2 hours
Rode 12 miles from C to F, which took them 12/20 or 3/5 hours
So it took the 3rd 4 also 2 3/5 hours to get from A to F

Edwin