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

$\displaystyle d= \left|\begin{array}{ccc}1 & 1 & 1 \\ 0 & -4 & 1 \\ 1 & -1 & 0\end{array}\right|$

$\displaystyle u= \left|\begin{array}{ccc}40 & 1 & 1 \\ 0 & -4 & 1 \\ -2 & -1 & 0\end{array}\right|$

$\displaystyle v= \left|\begin{array}{ccc}1 & 40 & 1 \\ 0 & 0 & 1 \\ 1 & -2 & 0\end{array}\right|$

$\displaystyle w= \left|\begin{array}{ccc}1 & 1 & 40 \\ 0 & -4 & 0 \\ 1 & -1 & -2\end{array}\right|$

Can you find those determinants by yourself?