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Math Help - Gaussian Elimination - 3 Simultaneous equations

  1. #1
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    Gaussian Elimination - 3 Simultaneous equations

    Hi all,

    Really, really struggling with this question.
    I've tried using a few examples but I cant seem to get narrow down 3 terms to just 1.
    I always end up with 2 letters/terms remaining.

    The 3 equations I have are:

    7i + 8ii + 7iii = 2 (Equation 1)
    6i + 7ii + 5iii = 8 (Equation 2)
    6i + 9ii + 6iii = 2 (Equation 3)

    So I need to work out the values for i, ii and iii.

    I tried using the 3 step procedure as described in a text book I have. But it doesn't seem to work.

    I hope someone can help.


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  2. #2
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    Re: Gaussian Elimination - 3 Simultaneous equations

    instead of i, ii and iii i will use a,b, and c. so our 3 equations become:

    7a + 8b + 7c = 2
    6a + 7b + 5c = 8
    6a + 9b + 6c = 2.

    subtract equation 2 from equation 3 to obtain:

    2b + c = -6 (*)

    now, we need another equation with just b and c in it, to see if we can eliminate another variable. since we want the "a" terms to cancel, multiply equation 1 by 6, and equation 2 by 7:

    42a + 48b + 42c = 12 (equation 1a)
    42a + 49b + 35c = 56 (equation 2a)

    then subtract equation 1a from equation 2a to get:

    b - 7c = 44 (**), and now we multiply (**) by 2 to get:

    2b - 14c = 88. subtract (*) from this, and we have:

    -15c = 94, so c = -94/15.

    using (*) we have:

    2b - 94/15 = -6
    b - 47/15 = -3
    b = -3 + 47/15 = -45/15 + 47/15 = 2/15

    and finally, using equation 1, we have:

    7a + 8(2/15) + 7(-94/15) = 2, so:

    a + 16/105 - 94/15 = 2/7 thus:

    a = 2/7 - 16/105 + 94/15 = 30/105 - 16/105 + 658/105 = 672/105 = 96/15 = 32/5.

    now let's see if our solution is correct (the first two times i did this, i made errors, so don't feel bad if you did, too):

    7(32/5) + 8(2/15) + 7(-94/15) = 224/5 + 16/15 - 658/15 = 4704/105 + 112/105 - 4606/105 = 210/105 = 2 (so equation 1 checks out).

    6(32/5) + 7(2/15) + 5(-94/15) = 192/5 + 14/15 - 94/3 = 576/15 + 14/15 - 470/15 = 120/15 = 8 (so equation 2 checks out).

    6(32/5) + 9(2/15) + 6(-94/15) = 192/5 + 6/5 - 188/5 = 10/5 = 2 (all three equations check out).

    so a = 32/5, b = 2/15, c = -94/15 is indeed the correct solution (with some rather ugly arithmetic attached).
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  3. #3
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    Re: Gaussian Elimination - 3 Simultaneous equations

    Thanks Deveno!

    That is perfect!!

    Going back over my original attempts I realise that NOT using fractions was a mistake! The decimal numbers got VERY messy.


    Thanks again, for taking the time to write it all out. I actually understand the process better now.

    Chris
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  4. #4
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    Re: Gaussian Elimination - 3 Simultaneous equations

    Hello, chrisa112!

    Didn't you say Gaussian elimination?


    \begin{array}{ccc|c}7a + 8b + 7c &=& 2 \\ 6a + 7b + 5c &=& 8 \\ 6a + 9b + 6c &=& 2 \end{array}

    We have: . \left|\begin{array}{ccc|c}7&8&7&2 \\ 6&7&5&8 \\ 6&9&6&2 \end{array}\right|


    \begin{array}{c}R_1-R_2 \\ \\ R_3-R_2 \end{array}\left|\begin{array}{ccc|c}1&1&2&\text{-}6 \\ 6&7&5&8 \\ 0&2&1&\text{-}6 \end{array}\right|

    \begin{array}{c} \\ R_2-6R_1 \\ \\ \end{array}\left|\begin{array}{ccc|c}1&1&2&\text{-}6 \\ 0&1&\text{-}7&44 \\ 0&2&1&\text{-}6 \end{array}\right|


    \begin{array}{c}R_1-R_2\ \\ R_3-2R_2 \end{array}\left|\begin{array}{ccc|c}1&0&9&\text{-}50 \\ 0&1&\text{-}7&44 \\ 0&0&15&\text{-}94 \end{array}\right|


    . . \begin{array}{c}\\ \\ \frac{1}{15}R_3 \end{array}\left|\begin{array}{ccc|c}1&0&9&\text{-}50 \\ 0&1&\text{-}7&44 \\ 0&0&1&\text{-}\frac{94}{15}\end{array}\right|


    \begin{array}{c}R_1 - 9R_3 \\ R_2 + 7R_3 \\ \\ \end{array}\left|\begin{array}{ccc|c}1&0&0&\frac{3  2}{5} \\ 0&1&0&\frac{2}{15} \\ 0&0&1&\text{-}\frac{94}{15} \end{array}\right|

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  5. #5
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    Re: Gaussian Elimination - 3 Simultaneous equations

    What Deveno did is also "Gaussian Elimination" as well as your row-reduction.
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  6. #6
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    Re: Gaussian Elimination - 3 Simultaneous equations

    Yes Deveno had solved it very smartly and this is a “Gaussian Elimination” even I got agreed with Hallsoflvy. Good job done by Deveno.
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