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Math Help - Simultaneous Equations

  1. #1
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    Simultaneous Equations

    Given that x and y satisfy the simultaneous equations
    mx + (m-1)y = 10
    (m-2)x + 3my = 20

    (a) If the equations have no unique solution, find the values of m. [1/2, -2]
    (b) If the equations have no solutions, find the values of m. [1/2]
    (c) If m =3, solve the simulataneous equations [x=2, y =2]

    Can this question be solved by NOT using the determinant method (i.e. b^2 - 4ac)
    For part a), this implies b^2 - 4ac = 0 (right?)
    For part b), this implies b^2 - 4ac = ?

    Thank you.
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  2. #2
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    The question is, do you know what you are trying to find the discriminant OF?
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  3. #3
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    Hi there,
    You're right! Didn't realize it - can you advise me on how to proceed?
    Thanks
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  4. #4
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    Solve equation 1 for \displaystyle y in terms of \displaystyle x, then substitute into equation 2.

    From there, the discriminant will tell you the values of \displaystyle m for which there are 0, 1, 2 solutions.
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  5. #5
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    In linear systems with paramater you usually use determinants
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  6. #6
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    Quote Originally Posted by mathfun View Post
    In linear systems with paramater you usually use determinants
    Oh crap, you're right. I could have sworn one of them was a quadratic equation ><
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  7. #7
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    I'm wondering...is there a reason you can't simply plug and play and then use simple algebra to solve for specific values? It doesn't seem that you have to do much more than that to solve the problems presented.
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  8. #8
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    Disregard my other posts...

    Write this equation in matrix form

    \displaystyle \left[\begin{matrix}m & m-1 \\ m - 2 & 3m\end{matrix}\right]\left[\begin{matrix}x \\ y\end{matrix}\right] = \left[\begin{matrix}10 \\ 20\end{matrix}\right]


    This matrix equation will have solutions where \displaystyle \left|\begin{matrix}m & m-1 \\ m-2 & 3m\end{matrix}\right| \neq 0.
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  9. #9
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    Thanks to all!!
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