Originally Posted by

**Jhevon** matrices are a pain to type. ok, first one of your equations are incorrect. let's work this guy out.

let x be the amount spent in 2001

let y be the amount spent in 2002

let z be the amount spent in 2003

since the total amount spent was 11 million, we have:

x + y + z = 11 mill

since the amount spent in 2003 was 3 times the amount spent in 2001, we have:

z = 3x

=> -3x + z = 0

since the amount spent in 2002 was $3 million less than the amount spent in 2003, we have:

y = z - 3 mill

=> y - z = -3 mill

so we have the system:

x + y + z = 11 mill ................(1)

-3x + z = 0 .........................(2)

y - z = -3 mill ......................(3)

converting this to matrices, we have:

[...x + y + z] =...[11 mill.]

|..-3x.....+ z| =..|....0 ...|

[.........y - z] =...[.-3 mill]

[1 ...1....1] [ x ] =...[11 mill.]

|-3...0....1| | y |=..|....0 ...|

[0....1...-1] [ z ] =...[.-3 mill]

To use Guass-Jordan elimination we need to have this in the form:

[1 ...1....1..|11 mill.]......[ x ]

|-3...0...1..|.0 ......|..= .[ y ]

[0....1..-1..|-3 mill.].......[ z ]

So we perform Guass-Jordan elimination to transform the matrix on the left to reduced row-echelon form. I'm getting tired of typing matrices, so i will just skip to the interesting part. if you are uncomfortable with the steps, say so, and i'll show them some time later on when i'm in the mood.

anyway, we end up with:

[1 ...0....0...|2 mill.]......[ x ]

|0....1....0...|.3 mill|..= .[ y ]

[0....0....1...|6 mill.].......[ z ]

Now we just read off our solutions from the last column of the matrix on the left.

so

x = 2 mill

y = 3 mill

z = 6 mill