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Nev3rforev3r
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Note: I didn't use the template because I feel it did not fit the question well enough.
This is concerning a system of linear equations in two variables where its constants in " ax+by=c " form show a geometric sequence, i.e. " nx + any = a2n ".
Another way of putting this is " y=(-1/a)x + a ". Note that in this second form, the constants are negative reciprocals of each other. I found that the general solution to any two equations following this form is (-ab,a+b) where a is the y-intercept of one equation and b is the y-intercept of the other.
If I graph a bunch of equations which follow one of the above general forms (like " y=(-1/5)x+5 " ) then they make a graph which looks like this: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs601.ash2/155353_1654251350058_1050271369_1853547_7385644_n.jpg"
As you can see, it forms a sort of sideways parabola. After some trial and error it appears to follow y^2=-4x.
Basically, I need to be able to mathematically show why that graphical pattern is there, preferably using that general solution (-ab,a+b) which I found.
Some notable information may include:
While solving for (-ab,a+b), I get to the point where y=(a+b)(a-b)/(a-b), indicating that there is a hole of some sort where a=b.
I think that where a=b there are infinite solutions to a set of any two equations, due to the fact that if a=b and the equations follow y=(-1/a)+a and y=(-1/b)+b, the lines are coincident.
All of these seems to be suggesting to me some sort of limit which is demonstrated by y^2=-4x or that all these lines are tangent to y^2=-4x, but I have been unable to prove this.
It has also occurred to me that it may be a separation of where a solution can be and where a solution cannot be.
Any hints? I don't really want just the answer, but if someone could tilt me in the right direction where I can figure it out mostly myself, it would be great.
Thank you all very much. I've worked hours and hours on this.
Big edit: I just tested a point (-1,0) in " nx + any = a2n ", yielding an answer of a= sqrt(-1)
WOAAAH
Another edit: If you take y2=-4x and input (-ba,a+b) it yields a2+b2=2ab. This is really freaky because of how close it is to the Pythagorean theorem and the law of cosines.
This is concerning a system of linear equations in two variables where its constants in " ax+by=c " form show a geometric sequence, i.e. " nx + any = a2n ".
Another way of putting this is " y=(-1/a)x + a ". Note that in this second form, the constants are negative reciprocals of each other. I found that the general solution to any two equations following this form is (-ab,a+b) where a is the y-intercept of one equation and b is the y-intercept of the other.
If I graph a bunch of equations which follow one of the above general forms (like " y=(-1/5)x+5 " ) then they make a graph which looks like this: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs601.ash2/155353_1654251350058_1050271369_1853547_7385644_n.jpg"
As you can see, it forms a sort of sideways parabola. After some trial and error it appears to follow y^2=-4x.
Basically, I need to be able to mathematically show why that graphical pattern is there, preferably using that general solution (-ab,a+b) which I found.
Some notable information may include:
While solving for (-ab,a+b), I get to the point where y=(a+b)(a-b)/(a-b), indicating that there is a hole of some sort where a=b.
I think that where a=b there are infinite solutions to a set of any two equations, due to the fact that if a=b and the equations follow y=(-1/a)+a and y=(-1/b)+b, the lines are coincident.
All of these seems to be suggesting to me some sort of limit which is demonstrated by y^2=-4x or that all these lines are tangent to y^2=-4x, but I have been unable to prove this.
It has also occurred to me that it may be a separation of where a solution can be and where a solution cannot be.
Any hints? I don't really want just the answer, but if someone could tilt me in the right direction where I can figure it out mostly myself, it would be great.
Thank you all very much. I've worked hours and hours on this.
Big edit: I just tested a point (-1,0) in " nx + any = a2n ", yielding an answer of a= sqrt(-1)
WOAAAH
Another edit: If you take y2=-4x and input (-ba,a+b) it yields a2+b2=2ab. This is really freaky because of how close it is to the Pythagorean theorem and the law of cosines.
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