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Math Help - System of two generic equations

  1. #1
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    System of two generic equations

    Hi,
    I am working on this problem

    Consider the following system of linear equations in x and y.

    ax + by = e

    cx + dy = f

    Under what conditions will the system have exactly one solution?

    I can come up with specific examples of when this happens but not a generic one just using the variables involved.
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  2. #2
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    Re: System of two generic equations

    The condition is ad-bc\ne0.

    Suppose ad-bc\ne0. Then x=\frac{de-bf}{ad-bc},y=\frac{af-ce}{ad-bc} is a solution. If x=x',y=y' is another solution, let X=\frac{de-bf}{ad-bc}-x',Y=\frac{af-ce}{ad-bc}-y'. This gives aX+bY=0=cX+dY, which yields X=0=Y, showing that the solution is unique.

    Now suppose ad-bc=0. We have 0=(ad-bc)x=de-bf and 0=(ad-bc)y=ce-af. Then either (1) de=bf and ce=af, in which case x and y can take any arbitrary values and so there are infinitely many solutions, or (2) de\ne bf or ce\ne af, in which case there is no solution in x and y.

    Hence the system has exactly one solution if and only if ad-bc\ne0.
    Last edited by Sylvia104; April 7th 2012 at 03:49 PM.
    Thanks from mash and bmoon123
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    Re: System of two generic equations

    I solved for x and y but didn't know of or think of the test for a second solution by creating the x' and y'. This is fantastically simple. I appreciate the help immensely.

    Also, unless I'm mistaken I think your last statement meant to say ad - bc not equal to 0, correct?
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  4. #4
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    Re: System of two generic equations

    Quote Originally Posted by bmoon123 View Post
    Also, unless I'm mistaken I think your last statement meant to say ad - bc not equal to 0, correct?
    Yes. Mistake corrected.
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    Re: System of two generic equations

    They are the equations of straight lines. Straight lines intersect at one point only unless they are parallel. So exactly one solution unless gradients equal, that is unless -b/a=-d/c, that is unless ad-bc=0
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    Re: System of two generic equations

    Quote Originally Posted by biffboy View Post
    They are the equations of straight lines. Straight lines intersect at one point only unless they are parallel. So exactly one solution unless gradients equal, that is unless -b/a=-d/c, that is unless ad-bc=0
    thanks, that makes sense. I think I tend to over think problems like this.

    My final answer:

    rewritten

    y = -(a/b)x + (e/b)

    y = -(c/d)x + (f/d)

    if a/b = c/d then the slopes of the lines are the same and they are parallel or coincident

    so a/b is not equal c/d is the condition needed for exactly one solution
    Last edited by bmoon123; April 8th 2012 at 07:35 AM.
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  7. #7
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    Re: System of two generic equations

    That's correct. We would usually finish it off by doing one extra step to say we require ad-bc not to be zero.
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