Results 1 to 3 of 3

Math Help - Systems of equations with arithmetic progression in coefficients

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    36

    Systems of equations with arithmetic progression in coefficients

    Hello guys,

    I am analyzing system of linear equations with coefficients that are part of arithmetic progression.

    Example of such system is

    \begin{array}{rcl}x+2y  & = & 3 \\ x-2y & = & -5\end{array}


    I should draw conclusions and prove them for 2x2 and 3x3 systems.

    2x2

    By plotting equation lines it is easy to conclude that all such 2x2 systems have unique solution [-1;2].

    By looking at a single equation
    ax+(a+k)y = a+2k \Leftrightarrow a(x+y)+ky=a+2k
    I have found that it will evaluate to true if
    ky=2k \Leftrightarrow y=2 \land x+y=1 \Leftrightarrow x=-1.

    Does this holds? I would appreciate very much if someone could provide real (rigor) proof.

    3x3

    I have tried few combinations of coefficients in matlab and got [0;-1;2] solution column vector few times, but I also got [NaN; Inf ; -Inf] in some cases. I do not have skills to prove this algebraically, any help?

    Thanks!
    Last edited by losm1; September 18th 2011 at 08:24 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Systems of equations with arithmetic progression in coefficients

    Quote Originally Posted by losm1 View Post
    Hello guys,

    I am analyzing system of linear equations with coefficients that are part of arithmetic progression.

    Example of such system is

    \begin{array}{rcl}x+2y  & = & 3 \\ x-2y & = & -5\end{array}


    I should draw conclusions and prove them for 2x2 and 3x3 systems.

    2x2

    By plotting equation lines it is easy to conclude that all such 2x2 systems have unique solution [-1;2].

    By looking at a single equation
    ax+(a+k)y = a+2k \Leftrightarrow a(x+y)+ky=a+2k
    I have found that it will evaluate to true if
    ky=2k \Leftrightarrow y=2 \land x+y=1 \Leftrightarrow x=-1.

    Does this holds? I would appreciate very much if someone could provide real (rigor) proof.

    3x3

    I have tried few combinations of coefficients in matlab and got [0;-1;2] solution column vector few times, but I also got [NaN; Inf ; -Inf] in some cases. I do not have skills to prove this algebraically, any help?

    Thanks!
    In the 2x2 case, it is correct that every line with equation of the form ax + (a+k)y = a+2k passes through the point (1,2). All you have to do to prove that is to plug the values x=1 and y=2 into the equation and check that the equation is satisfied. If you have two equations of that form, they will usually have that as their unique solution. However, there is an exceptional case, namely when the two lines coincide. For example, the equations

    x+2y=3
    2x+4y=6

    have infinitely many solutions, because every point on the first line is also on the second line.

    In the 3x3 case, an equation of the form ax + (a+k)y + (a+2k)z = a+3k represents a plane in three-dimensional space. It will always contain the whole of the line (-2,3,0) + t(1,-2,1). Again, you can verify that by plugging x=-2+t, y=3-2t and z=t into the equation and checking that it is satisfied. In particular, if you put t=2 then you see that the point (0,-1,2) lies in the plane.

    So if you take any number of planes of that form, their intersection will always contain that line. For two or more planes, the intersection will normally be exactly equal to that line. But as in the 2x2 case, there is an exceptional case that arises if all the equations are multiples of each other and therefore represent the same plane. In that case, every point of the plane will give a solution to the equations.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    36

    Re: Systems of equations with arithmetic progression in coefficients

    Can someone help me prove that plane ax + (a+k)y + (a+2k)z = a+3k always contains line that Opalg described?

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Arithmetic progression
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 26th 2011, 03:13 PM
  2. Arithmetic Progression
    Posted in the Algebra Forum
    Replies: 5
    Last Post: January 24th 2010, 06:05 AM
  3. Arithmetic Progression or Arithmetic Series Problem
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: October 8th 2009, 12:36 AM
  4. Replies: 8
    Last Post: March 23rd 2009, 07:26 AM
  5. arithmetic progression
    Posted in the Algebra Forum
    Replies: 9
    Last Post: November 10th 2008, 03:57 AM

Search Tags


/mathhelpforum @mathhelpforum