Math Help - Solve matrices using Gaussian elimination or...

Gauss-Jordan elimination.

This is a word problem and I'm not very good at these, so please help.

eAuction.com spent a total of $11 million on advertising in fiscal years 2001, 2002, and 2003. The amount spent in 2003 was 3 times the amount spent in 2001. The amount spent in 2002 was $3 million less than the amount spent in 2003. How much was spent on advertising each year?

I'm not sure if this is how I should proceed but this is a far as my brain would stretch...

x = 2001
y = 2002
z = 2003

x + y + z = $11 million

I'm rambling here:

3z - x = ?

z - 3,000,000 = y

How do you use Gaussian elimination or Gauss-Jordan elimination to solve this? Thanks in advance.

Originally Posted by lilrhino

Gauss-Jordan elimination.

This is a word problem and I'm not very good at these, so please help.

eAuction.com spent a total of $11 million on advertising in fiscal years 2001, 2002, and 2003. The amount spent in 2003 was 3 times the amount spent in 2001. The amount spent in 2002 was $3 million less than the amount spent in 2003. How much was spent on advertising each year?

I'm not sure if this is how I should proceed but this is a far as my brain would stretch...

x = 2001
y = 2002
z = 2003

x + y + z = $11 million

I'm rambling here:

3z - x = ?

z - 3,000,000 = y

How do you use Gaussian elimination or Gauss-Jordan elimination to solve this? Thanks in advance.

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

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

All that I can say is WOW! Your explanation is so thorough that I'll definitely be able to understand it and use it to solve other problems like this. Thanks so much!

Originally Posted by lilrhino

All that I can say is WOW! Your explanation is so thorough that I'll definitely be able to understand it and use it to solve other problems like this. Thanks so much!

really, i actually didn't explain anything, i don't think. i left out the hardest part of the problem--actually doing the Guass-Jordan Elimination. do you have problems with doing the elimination?

Originally Posted by Jhevon

really, i actually didn't explain anything, i don't think. i left out the hardest part of the problem--actually doing the Guass-Jordan Elimination. do you have problems with doing the elimination?

No, I don't have a problem with doing the elimination, I know how to do it. I just didn't know how to convert the word problem into 3 systems of equations. Go figure...