Expert: Steve Holleran - 11/29/2007

Question
Write the following problems as a system of equations, and then write that system as matrix equation. Find and use an inverse matrix to solve the problem.

a. The sum of three numbers is 40. One number is five more than a second number. It is also twice the third. Find the numbers.

b. A Boeing 727 traveled 560 mph with the wind and 480 mph against the same wind. Find the speed of the plane in still air and the speed of the wind.

Answer
Hi Alex,

Okay, for the first one, we have these equations:

   x + y + z = 40  ---------->  x + y + z = 40

   x = y + 5       ---------->  x - y     = 5

   x = 2 z         ---------->  x      -2z = 0

As a matrix equation, its:

      A         *     B     =   C

  [ 1  1  1     *  [ x      = [ 40

    1  -1  0         y          5

    1  0  -2]        z]         20]

Then B = A^-1 * C = [ 22

                     17

                      1]

On b, I've never seen these done using matrices.  Anyway, there is a problem here.  Is the time equal in each case?

If you let p = the plane's speed, and w = the wind, then the equations you have are:

    560 = (p + w) * T --------> pT + wT = 560

    480 = (p - w) * T --------> pT - wT = 480

But since you don't know the plane's speed or the wind's speed, and there is no information about the times, I don't see how you set up a meaningful matrix.  You need to know if the times are equal.  If they are, then solve each for T:

    T = 560 / (p+w)      and t = 480 / (p - w)

then       560 / (p+w)  = 480 / (p-w)

and         560p - 560w = 480p + 480w

but all this tells you is that p = 130w... I think more infor is needed here.

Hope this helps a little.
Steve