math:3x3numbersystems
Differences
This shows you the differences between two versions of the page.
Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
math:3x3numbersystems [2010/03/22 09:29] – mcooper6 | math:3x3numbersystems [2010/03/22 22:54] (current) – mcooper6 | ||
---|---|---|---|
Line 1: | Line 1: | ||
+ | ~~DISCUSSION|3x3 Number Discussion~~ | ||
+ | <WRAP left 40%> | ||
+ | ~~TOC~~ | ||
+ | </ | ||
+ | // | ||
+ | // | ||
+ | <WRAP centeralign 100% bigger> | ||
+ | <WRAP bigger fgred> | ||
+ | <WRAP muchbigger> | ||
+ | An Exploration of Methods</ | ||
+ | |||
+ | <WRAP left 40%> | ||
+ | <WRAP right bigger fgred bgwhite> | ||
+ | |||
+ | \\ | ||
+ | Eq1-> < | ||
+ | \\ | ||
+ | Eq2-> < | ||
+ | \\ | ||
+ | Eq3-> < | ||
+ | \\ | ||
+ | </ | ||
+ | </ | ||
+ | |||
+ | <WRAP clear></ | ||
+ | |||
+ | =====Addition Method===== | ||
+ | |||
+ | ====Step 1:===== | ||
+ | |||
+ | Use the addition method to remove one of the variables from the equation. | ||
+ | This step may, in some situations, cause two variables to drop out if you're lucky. | ||
+ | |||
+ | <wrap bigger fgred> | ||
+ | \\ | ||
+ | \\ | ||
+ | <WRAP indent big> | ||
+ | |||
+ | < | ||
+ | \begin{array}{3} | ||
+ | -1(3x + y - 3z = 11)\\ | ||
+ | \underline{+ 3x - 3y + 2z = 11}\\ | ||
+ | -4y + 5z = 0 \\ | ||
+ | \end{array}\\ | ||
+ | </ | ||
+ | |||
+ | </ | ||
+ | |||
+ | <WRAP info bgwhite 50%> | ||
+ | At first, it may seem that you're only multiplying one equation by <wrap fgred> | ||
+ | </ | ||
+ | |||
+ | ====Step 2:==== | ||
+ | |||
+ | Use the addition method to remove the same variable from a different set of equations. | ||
+ | |||
+ | <WRAP bigger fgred> | ||
+ | (-5)Eq1 + Eq3 | ||
+ | </ | ||
+ | \\ | ||
+ | <WRAP indent big> | ||
+ | |||
+ | < | ||
+ | \begin{array}{3} | ||
+ | -5(3x - 3y + 2z = 11)\\ | ||
+ | \underline{+ 15x - | ||
+ | 14y - 12z = -55\\ | ||
+ | .\end{array}\\ | ||
+ | </ | ||
+ | |||
+ | </ | ||
+ | |||
+ | ====Step 3 (determinates of a 2x2 matrix)==== | ||
+ | Two new equations have been introduced. | ||
+ | <WRAP bigger indent fgred> | ||
+ | Eq4 -> < | ||
+ | \\ | ||
+ | \\ | ||
+ | Eq5 -> < | ||
+ | </ | ||
+ | \\ | ||
+ | \\ | ||
+ | Create 3 2 x 2 martices as follows: | ||
+ | *The first row of Matrix < | ||
+ | *The second row of Matrix < | ||
+ | *The first row of Matrix < | ||
+ | *The second row of Matrix < | ||
+ | *The first row of Matrix < | ||
+ | *The second row of Matrix < | ||
+ | \\ | ||
+ | \\ | ||
+ | <WRAP indent> | ||
+ | < | ||
+ | y_1=-4 , y_2=14 | ||
+ | </ | ||
+ | \\ | ||
+ | \\ | ||
+ | < | ||
+ | z_1=5 , z_2=-12 | ||
+ | </ | ||
+ | \\ | ||
+ | \\ | ||
+ | < | ||
+ | C_1=0 , C_2=-55 | ||
+ | </ | ||
+ | \\ | ||
+ | \\ | ||
+ | |||
+ | < | ||
+ | D | ||
+ | = \left| | ||
+ | \begin{array}{ccc} | ||
+ | y_1 ... z_1\\ | ||
+ | \vdots \ddots \vdots\\ | ||
+ | y_2 ... z_2 \end{array} | ||
+ | \right| | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | D_y | ||
+ | = \left| | ||
+ | \begin{array}{ccc} | ||
+ | C_1 ... z_1\\ | ||
+ | \vdots \ddots \vdots\\ | ||
+ | C_2 ... z_2\end{array} | ||
+ | \right| \to y = \frac{D_y}{D} \to y = \frac{(0)(-12)-(-55)(5)}{(-4)(-12)-(14)(5)} \to y = \frac{25}{2} | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | D_z | ||
+ | = \left| | ||
+ | \begin{array}{ccc} | ||
+ | y_1 ... C_1\\ | ||
+ | \vdots \ddots \vdots\\ | ||
+ | y_2 ... C_2\end{array} | ||
+ | \right| \to z = \frac{D_z}{D} \to y = \frac{(-4)(-55)-(14)(0)}{(-4)(-12)-(14)(5)} \to z = -10 | ||
+ | </ | ||
+ | |||
+ | </ | ||
+ | |||
+ | ====Step 4: Solve it==== | ||
+ | Now that < | ||
+ | |||
+ | < | ||
+ | \\ | ||
+ | y = \frac{25}{2} | ||
+ | \\ | ||
+ | \\ | ||
+ | z = -10 | ||
+ | \\ | ||
+ | </ | ||
+ | < | ||
+ | |||
+ | < | ||
+ | \\ | ||
+ | 3x - 3y + 2z = 11 \to 3x - 3(\frac{25}{2}) + 2(-10) = 11 \to x = \frac{3(\frac{25}{2}) + 2(-10) - 11}{3} | ||
+ | \\ | ||
+ | \\ | ||
+ | x = \frac{-13}{6} | ||
+ | </ | ||
+ | |||
+ | </ | ||
+ | |||
+ | =====Determinants of the Third Order===== | ||
+ | <WRAP indend fgred bigger> | ||
+ | < | ||
+ | \\ | ||
+ | < | ||
+ | \\ | ||
+ | < | ||
+ | |||
+ | <WRAP info bgwhite> | ||
+ | Equations given in general form can be solved using determinants with the following formulas: | ||
+ | \\ | ||
+ | \\ | ||
+ | < | ||
+ | \\ | ||
+ | x= \frac{d_1b_2c_2 + d_3b_1c_2 + d_2b_3c_1 - d_3b_2c_1 - d_1b_3c_2 - d_2b_1c_3}{a_1b_2c_3 + a_3b_1c_2 + a_2b_3c_1 - a_3b_2c_1 - a_1b_2c_3 - a_2b_1c_3} | ||
+ | \\ | ||
+ | \\ | ||
+ | y= \frac{a_1d_2c_3 + a_3d_1c_2 + a_2d_3c_1 - a_3d_2c_1 - a_1d_3c_2 - a_2d_1c_3}{a_1b_2c_3 + a_3b_1c_2 + a_2b_3c_1 - a_3b_2c_1 - a_1b_2c_3 - a_2b_1c_3} | ||
+ | \\ | ||
+ | \\ | ||
+ | z= \frac{a_1b_2d_3 + a_3b_1d_2 + a_2b_3d_1 - a_3b_2d_1 - a_1b_3d_2 - a_2b_1d_3}{a_1b_2c_3 + a_3b_1c_2 + a_2b_3c_1 - a_3b_2c_1 - a_1b_2c_3 - a_2b_1c_3} | ||
+ | \\ | ||
+ | </ | ||
+ | </ |