Expert: Sherry Wallin - 8/6/2009

Question
-0.8x+0.3y+0.2z=0
0.5x-0.4y+0.3z=0
0.3x+0.1y-0.5z=0
where: x+y+z=1

how do i find the values of x, y, and z? please detail the process

Answer
Jo Anne, there are a number of ways to find the solution to this system of equations. First I would like to know if you have worked with matrices before?  Also, you can multiply all the equations by 10 to get rid of the decimals:
-8x+3y+2z = 0  (1)
5x -4y+3z = 0  (2)
3x+y-5z = 0    (3)

Also knowing that x+y+z = 1 is good to know (but isn't even needed for this problem)

Multiply eqn (2) by 3 and eqn (3) by -5 getting
15x -12y+9z = 0  (2a)
-15x-5y+25z = 0  (3a)  add these getting
-17y + 34z = 0   (4)

Mulitply eqn (1) by 3 and eqn (3) by 8 getting
-24x+9y+6z = 0   (1a)
24x+8y-40z = 0   (3b)
17y - 34z = 0   (5)

Note (4) and (5) are opposites of each other which basically means you have a dependent system, i.e., eqn 2 and eqn 3 are multiples of one another. You can solve for x in terms of y from eqn 1 getting x = (3/8)y +(1/4)z, this will generate a gazillion (infinite) number of solutions.

Math Prof