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Math Help - Cramer's Rule

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by magentarita View Post
    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?
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  3. #3
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    yes...

    Quote Originally Posted by HallsofIvy View Post
    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.
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  4. #4
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    I do problems like that while my boss isn't looking!
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  5. #5
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    me too...

    Quote Originally Posted by HallsofIvy View Post
    I do problems like that while my boss isn't looking!
    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.
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  6. #6
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    Quote Originally Posted by magentarita View Post
    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.
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  7. #7
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    Thanks...

    Quote Originally Posted by janvdl View Post
    With determination like that, you will succeed.
    Thank you for believing in me.
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