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Math Help - Simultaneous Equation

  1. #1
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    Simultaneous Equation

    1) Solve simultaneously:

    3u+v-4w=-4
    u-2v+7w=-7
    4u+3v-w=9
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  2. #2
    Super Member angel.white's Avatar
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    Quote Originally Posted by acevipa View Post
    1) Solve simultaneously:

    3u+v-4w=-4
    u-2v+7w=-7
    4u+3v-w=9
    Initial matrix
    \begin{array}{|rrrcr|rcrrrcrl}<br />
3u & +v & -4w & = & -4 &  &  &  &  &  &  &  & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 &  &  &  &  &  &  &  & \\<br />
\end{array}

    --------------------

    Subtract 3 line twos from line 1
    \begin{array}{|rrrcr|rcrrrcrl}<br />
3u & +v & -4w & = & -4 &  -3& ( & u & -2v & +7w & = & -7 & )\\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 &  &  &  &  &  &  &  & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
3u & +v & -4w & = & -4 &  -& ( & 3u & -6v & +21w & = & -21 & )\\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 &  &  &  &  &  &  &  & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 &  &  &  &  &  &  &  & \\<br />
\end{array}

    --------------------

    Subtract 4 line twos from line 3
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 & -4 & ( & u & -2v & +7w & = & -7 & )\\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
4u & +3v & -w & = & 9 & - & ( & 4u & -8v & +28w & = & -28 & )\\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & 11v & -29w & = & 37 & & & & & & & & \\<br />
\end{array}

    --------------------

    Subtract 11/7 line ones from line 3
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & 11v & -29w & = & 37 & -11/7 & ( &  & 7v & -25w & = & 17 & )\\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & 11v & -29w & = & 37 & - & ( &  & 11v & -(275/7)w & = & 187/7 & )\\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & & (72/7)w & = & 72/7 & & & & & & & & \\<br />
\end{array}

    --------------------

    Multiply line 3 by 7/72
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & & (72/7)w & = & 72/7 & * & ( & 7/72 & = & 7/72 & ) & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25w & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7w & = & -7 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Substitute the value of w into lines 1 and 2
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25(1) & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7(1) & = & -7 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25 & = & 17 &  &  &  &  &  &  & & \\<br />
u & -2v & +7 & = & -7 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Add 25 to line1 and subtract 7 from line2
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & -25 & = & 17 & + & ( & 25 & = & 25 & ) & & \\<br />
u & -2v & +7 & = & -7 & - & ( & 7 & = & 7 & ) &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & & = & 42 &  &  &  &  &  &  & & \\<br />
u & -2v & & = & -14 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Multiply line one by 1/7
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & 7v & & = & 42 & * & ( & 1/7 & = & 1/7 & ) & & \\<br />
u & -2v & & = & -14 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & v & & = & 6 & & & & & & & & \\<br />
u & -2v & & = & -14 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Substitute the value of v into line 2
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & v & & = & 6 & & & & & & & & \\<br />
u & -2(6) & & = & -14 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & v & & = & 6 & & & & & & & & \\<br />
u & -12 & & = & -14 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Add 12 to line2
    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & v & & = & 6 & & & & & & & & \\<br />
u & -12 & & = & -14 & + & ( & 12 & = & 12 & ) &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    \begin{array}{|rrrcr|rcrrrcrl}<br />
 & v & & = & 6 & & & & & & & & \\<br />
u & & & = & -2 &  &  &  &  &  &  &  & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}

    --------------------

    Switch lines 1 and 2
    \begin{array}{|rrrcr|rcrrrcrl}<br />
u & & & = & -2 &  &  &  &  &  &  &  & \\<br />
 & v & & = & 6 & & & & & & & & \\<br />
 & & w & = & 1 &  & & & & & & \\<br />
\end{array}
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  3. #3
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    Beautiful latex, angel.white

    An alternative approach that avoids explicitly using matrices:

    Quote Originally Posted by acevipa View Post
    1) Solve simultaneously:

    3u+v-4w=-4 .... (1)

    u-2v+7w=-7 .... (2)

    4u+3v-w=9 .... (3)

    Edited by Mr F.
    (1) - 3 x (2): 7v - 25w = 17 .... (A)

    (3) - 4 x (2): 11v - 29w = 37 .... (B)

    Now solve (A) and (B) in the usual way (I'd suggest elimination method): v = 6, w = 1.

    Now sub v = 6 and w = 1 into either (1), (2) or (3).
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  4. #4
    Super Member angel.white's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Beautiful latex, angel.white

    An alternative approach that avoids explicitly using matrices:



    (1) - 3 x (2): 7v - 25w = 17 .... (A)

    (3) - 4 x (2): 11v - 29w = 37 .... (B)

    Now solve (A) and (B) in the usual way (I'd suggest elimination method): v = 6, w = 1.

    Now sub v = 6 and w = 1 into either (1), (2) or (3).
    Thank you My next goal is to figure out how to do long division with them. I've tried twice already with unimpressive results, but I can probably pull it off if I nest them enough.
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