Assume that, on a certain day, 495 passengers want to fly from Honolulu (HNL) to New York (JFK); 605 passengers wish to fly from HNL to Los Angeles (LAX); and 1100 passengers want to fly from LAX to JFK.
All the passengers travelling to and from LAX have direct flights, but only some of the passengers among HNL and JFK are flying direct, while the remainder have a layover in LAX.
Let x be the actual number of passengers who fly from HNL to JFK, let y be the total number who fly from HNL to LAX, and let z be the total number who fly from LAX to JFK.
Derive the linear system of two equations in the three unknowns (x, y, z). Use Gauss-Jordan elimination to determine the solution set.
Determine the matrix of transformation for the orthogonal projection onto the line L which passes through the origin and is in the direction Û=(3/13, 4/13, 12/13). Determine the rank of this matrix and describe what it tells you about the possible solutions to the system projL (x) = b for some suitable vector b.