# Simultaneous Equation

• Feb 11th 2008, 11:13 AM
acevipa
Simultaneous Equation
1) Solve simultaneously:

$\displaystyle 3u+v-4w=-4$
$\displaystyle u-2v+7w=-7$
$\displaystyle 4u+3v-w=9$
• Feb 11th 2008, 11:45 AM
angel.white
Quote:

Originally Posted by acevipa
1) Solve simultaneously:

$\displaystyle 3u+v-4w=-4$
$\displaystyle u-2v+7w=-7$
$\displaystyle 4u+3v-w=9$

Initial matrix
$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} 3u & +v & -4w & = & -4 & & & & & & & & \\ u & -2v & +7w & = & -7 & & & & & & & & \\ 4u & +3v & -w & = & 9 & & & & & & & & \\ \end{array}$

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

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & 7v & -25w & = & 17 & & & & & & & & \\ u & -2v & +7w & = & -7 & & & & & & & & \\ 4u & +3v & -w & = & 9 & & & & & & & & \\ \end{array}$

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

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & 7v & -25w & = & 17 & & & & & & & & \\ u & -2v & +7w & = & -7 & & & & & & & & \\ & 11v & -29w & = & 37 & & & & & & & & \\ \end{array}$

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

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

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

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & 7v & -25w & = & 17 & & & & & & & & \\ u & -2v & +7w & = & -7 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & 7v & -25 & = & 17 & & & & & & & & \\ u & -2v & +7 & = & -7 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & 7v & & = & 42 & & & & & & & & \\ u & -2v & & = & -14 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & v & & = & 6 & & & & & & & & \\ u & -2v & & = & -14 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & v & & = & 6 & & & & & & & & \\ u & -12 & & = & -14 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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

$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} & v & & = & 6 & & & & & & & & \\ u & & & = & -2 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$

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Switch lines 1 and 2
$\displaystyle \begin{array}{|rrrcr|rcrrrcrl} u & & & = & -2 & & & & & & & & \\ & v & & = & 6 & & & & & & & & \\ & & w & = & 1 & & & & & & & \\ \end{array}$
• Feb 12th 2008, 12:38 AM
mr fantastic
Beautiful latex, angel.white (Clapping)

An alternative approach that avoids explicitly using matrices:

Quote:

Originally Posted by acevipa
1) Solve simultaneously:

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

$\displaystyle u-2v+7w=-7$ .... (2)

$\displaystyle 4u+3v-w=9$ .... (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).
• Feb 12th 2008, 12:56 AM
angel.white
Quote:

Originally Posted by mr fantastic
Beautiful latex, angel.white (Clapping)

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.