# Cramer's Rule

• Dec 29th 2008, 10:02 PM
magentarita
Cramer's Rule
As a participant in your school's community service project, you volunteer a total of 40 hours over the course of the school year. Your volunteer hours include serving at a soup kitchen, picking up trash at several local parks, and collecting toys for needy children. You spend 4 times as many hours collecting toys as picking up trash, and 2 hours less serving at the soup kitchen than picking up trash.

Solve this system of equations using Cramer's Rule.
• Dec 30th 2008, 03:24 AM
HallsofIvy
Quote:

Originally Posted by magentarita
As a participant in your school's community service project, you volunteer a total of 40 hours over the course of the school year. Your volunteer hours include serving at a soup kitchen, picking up trash at several local parks, and collecting toys for needy children. You spend 4 times as many hours collecting toys as picking up trash, and 2 hours less serving at the soup kitchen than picking up trash.

Solve this system of equations using Cramer's Rule.

First, of course, you need a system of equations! (In fact, in my opinion, that's a badly stated problem. What you are really asked to do is find how many hours you spend doing each of three different kinds of things You shouldn't be asked to "solve the system of equation" when no system of equations has been given!). Let "s" be the number of hours you spend serving at the soup kitchen, "p" be the number of hours you spend picking up trash, and "c" be the number of hours you spend collecting toys.

Now translate each sentence into an equation:
" you volunteer a total of 40 hours" s+ p+ c= 40.

"You spend 4 times as many hours collecting toys as picking up trash"
c= 4p.

"You spend 2 hours less serving at the soup kitchen than picking up trash"
s= p- 2.

So your three equations can be written
s+ p+ c= 40
-4p+ c= 0
s- p= -2

Now, Cramer's rule says that the solution can be written s= u/d, p= v/d, and c= w/d where d is the determinant formed from the coefficients and u, v, w are the same determinant but with the first, second, and third columns, respectively, replaced by the numbers on the right hand side of the equations.
That is
$d= \left|\begin{array}{ccc}1 & 1 & 1 \\ 0 & -4 & 1 \\ 1 & -1 & 0\end{array}\right|$
$u= \left|\begin{array}{ccc}40 & 1 & 1 \\ 0 & -4 & 1 \\ -2 & -1 & 0\end{array}\right|$
$v= \left|\begin{array}{ccc}1 & 40 & 1 \\ 0 & 0 & 1 \\ 1 & -2 & 0\end{array}\right|$
$w= \left|\begin{array}{ccc}1 & 1 & 40 \\ 0 & -4 & 0 \\ 1 & -1 & -2\end{array}\right|$

Can you find those determinants by yourself?
• Dec 30th 2008, 08:58 PM
magentarita
yes...
Quote:

Originally Posted by HallsofIvy
First, of course, you need a system of equations! (In fact, in my opinion, that's a badly stated problem. What you are really asked to do is find how many hours you spend doing each of three different kinds of things You shouldn't be asked to "solve the system of equation" when no system of equations has been given!). Let "s" be the number of hours you spend serving at the soup kitchen, "p" be the number of hours you spend picking up trash, and "c" be the number of hours you spend collecting toys.

Now translate each sentence into an equation:
" you volunteer a total of 40 hours" s+ p+ c= 40.

"You spend 4 times as many hours collecting toys as picking up trash"
c= 4p.

"You spend 2 hours less serving at the soup kitchen than picking up trash"
s= p- 2.

So your three equations can be written
s+ p+ c= 40
-4p+ c= 0
s- p= -2

Now, Cramer's rule says that the solution can be written s= u/d, p= v/d, and c= w/d where d is the determinant formed from the coefficients and u, v, w are the same determinant but with the first, second, and third columns, respectively, replaced by the numbers on the right hand side of the equations.
That is
$d= \left|\begin{array}{ccc}1 & 1 & 1 \\ 0 & -4 & 1 \\ 1 & -1 & 0\end{array}\right|$
$u= \left|\begin{array}{ccc}40 & 1 & 1 \\ 0 & -4 & 1 \\ -2 & -1 & 0\end{array}\right|$
$v= \left|\begin{array}{ccc}1 & 40 & 1 \\ 0 & 0 & 1 \\ 1 & -2 & 0\end{array}\right|$
$w= \left|\begin{array}{ccc}1 & 1 & 40 \\ 0 & -4 & 0 \\ 1 & -1 & -2\end{array}\right|$

Can you find those determinants by yourself?

Yes, I can find the determinants. I'll do that on my next day off.
• Dec 31st 2008, 04:56 AM
HallsofIvy
I do problems like that while my boss isn't looking!(Rofl)
• Jan 1st 2009, 12:29 PM
magentarita
me too...
Quote:

Originally Posted by HallsofIvy
I do problems like that while my boss isn't looking!(Rofl)

Believe it or not, I take math sheets to work and work out math questions in the bathroom and on my lunch break. I want to master this stuff.
• Jan 1st 2009, 12:33 PM
janvdl
Quote:

Originally Posted by magentarita
Believe it or not, I take math sheets to work and work out math questions in the bathroom and on my lunch break. I want to master this stuff.

With determination like that, you will succeed. (Clapping)
• Jan 3rd 2009, 07:43 AM
magentarita
Thanks...
Quote:

Originally Posted by janvdl
With determination like that, you will succeed. (Clapping)

Thank you for believing in me.