Results 1 to 3 of 3

Math Help - RREF for 3-Variable Equations with Geometric Ratios

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    14

    RREF for 3-Variable Equations with Geometric Ratios

    Hello everyone, I'll start off my first post by explaining a few things.

    The equations used in this matrix all have their coefficients related by a common ratio, for example:

    8x+4y+2z=1 all have a ratio of 2 and so forth.

    When I have three of such equations I find that the there is always a definite solution to these equations. With its appearance being that of an identity matrix I was wondering if there were any pattern related to the solutions as I found one for a 2x2 matrix.

    Here are my algebraic expressions:

    ax+(ar)y+(ar^2)z=ar^3
    bx+(bn)y+(bn^2)z=bn^3
    jx+(jk)y+(jk^2)z=jk^3

    where a,b and j are not equal, and r/n/k are ratios respectively.

    I put this into a matrix in its condensed form

    Could anybody help me with using the Guassian method of elimination of help me obtain a resulting solution where it will look like a 3x3 identity matrix.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,905
    Thanks
    765
    Hello, chaosier!

    \begin{array}{cccc}ax+(ap)y+(ap^2)z &=& ap^3 & [1] \\<br />
bx+(bq)y+(bq^2)z &=& bq^3 & [2] \\<br />
cx+(cr)y+(cr^2)z &=& cr^3 & [3] \end{array}
    \begin{array}{cccccc}\text{Divide [1] by }a\!: & x + py + p^2z &=& p^3 \\ \text{Divide [2] by }b\!: &  x + qy +q^2z &=& q^2 \\ \text{Divide [3] by }c\!: &  x + ry + r^2z &=& r^3 \end{array}


    \text{We have: }\;\left[\begin{array}{ccc|c} 1 & p & p^2 & p^3 \\<br />
1 & q & q^2 & q^3 \\ 1 & r & r^2 & r^3 \end{array}\right]


    \begin{array}{c} \\ R_2-R_1 \\ R_3-R_1\end{array}<br />
\left[\begin{array}{ccc|c}1 & p & p^2 & p^3 \\<br />
0 & q-p & q^2-p^2 & q^3-p^3 \\<br />
0 & r-p & r^2-o^2 & r^3-p^3 \end{array}\right]


    . . \begin{array}{ccc} . \\ . \\ \frac{1}{q-p}R_2 \\ \frac{1}{r-p}R_3 \end{array} . \left[ \begin{array}{ccc|c} 1 & p & p^2 & p^3 \\<br />
0 & 1 & q+r & q^2+qp + p^2 \\<br />
0 & 1 & r+p & r^2+rp + p^2 \end{array} \right]


    \begin{array}{c} R_1 - p\!\cdot\!R_2 \\ \\ R_3 - R_2 \end{array} <br />
\left[\begin{array}{ccc|c}1 & 0 & -pq & -pq(p+q) \\<br />
0 & 1 & p+q & p^2 + pq + q^2 \\<br />
0 & 0 & r-q & (r^2-q^2) + p(r-q) \end{array}\right]


    . . \begin{array}{ccc} \\ \\ \frac{1}{r-q}R_3 \end{array}<br />
\left[\begin{array}{ccc|c} 1 & 0 & -pq & -pq(p+q) \\<br />
0 & 1 & p+q & p^2 + pq + q^2 \\<br />
0 & 0 & 1 & p+q+r \end{array}\right]


    \begin{array}{c}R_1 + pq\!\cdot\!R_3 \\ R_2 - (p+q)R_3 \\ \end{array}<br />
\left[\begin{array}{ccc|c}1 & 0 & 0 & pqr \\<br />
0 & 1 & 0 & -(pq+qr+pr) \\<br />
0 & 0 & 1 & p+q+r \end{array}\right]


    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    14
    Thanks a lot, by the way how do you get that straight line separating the 3x3 from the 3x1 in MathType
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Geometric Random Variable
    Posted in the Statistics Forum
    Replies: 6
    Last Post: November 8th 2010, 12:15 PM
  2. Geometric Random Variable.
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 22nd 2010, 09:52 PM
  3. Geometric random variable
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: August 11th 2009, 02:48 AM
  4. Geometric equations with variable terms
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: June 4th 2009, 07:57 PM
  5. Ratios/equations
    Posted in the Algebra Forum
    Replies: 4
    Last Post: January 23rd 2009, 03:36 AM

Search Tags


/mathhelpforum @mathhelpforum