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Math Help - system of equations: no solution, unique solution, infinitely many solutions

  1. #1
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    system of equations: no solution, unique solution, infinitely many solutions

    Hey guys,

    I have to solve the following problem:

    Let the following systems of equations be given:
    i)
    2x 1 − 3x 2 + x 3 = 4
    x 1 − 4x 2 − x 3 = 2
    − 3x 1 + tx 2 − 2x 3 = − 3
    ii)
    x 2 − x 3 = 2
    − x 1 + 2x 2 − 4x 3 = 3
    x 1 − x 2 + 3x 3 = t + 1

    For which values of t does the system have
    a) a unique solution? Find this solution.
    b) infinitely many solutions? Find the solution set.
    c) no solution?

    So I started with the first one and get
    x2 = -7/(8+t)

    so I guess c) if t=8 the system has no solution.

    For the others I just don't know how to proceed... Thanks for any help. :)
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  2. #2
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    Re: system of equations: no solution, unique solution, infinitely many solutions

    system of equations: no solution, unique solution, infinitely many solutions-linear-equations.png
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  3. #3
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    Re: system of equations: no solution, unique solution, infinitely many solutions

    Thank you ibdutt. This is very helpful but using this method in this special case I get:

    for no solution: 5/(8+t) = 3/0
    for infinitely many solutions: 5/(8+t) = 3/0 = 0/7
    for unique solution: 5/(8+t) /= 3/0

    ....??
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  4. #4
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    Re: system of equations: no solution, unique solution, infinitely many solutions

    Quote Originally Posted by pesto View Post
    Thank you ibdutt. This is very helpful but using this method in this special case I get:

    for no solution: 5/(8+t) = 3/0
    which is the same as 0(5)= 2(8+ t). What does that tell you?

    for infinitely many solutions: 5/(8+t) = 3/0 = 0/7
    Which is the same as 3(7)= 0. What does that tell you?

    for unique solution: 5/(8+t) /= 3/0
    Which is the same as 3(8+t) is NOT equal to 0. What does that tell you?

    ....??
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  5. #5
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    Re: system of equations: no solution, unique solution, infinitely many solutions

    well maybe that there is no solution for t = -8 and a unique solution for t /= -8 and that there are no infinitely many solutions..?
    But if all t /= -8 have a unique solution I would think that there are infinitely many?!
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