How many equations

Discussion in 'Math' started by studiot, Apr 23, 2015.

  1. studiot

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    How many equations would you say this expression represents 1,2, 3 or more?

     \frac{x- x_{1} }{ x_{2}- x_{1}  } =\frac{y- y_{1} }{ y_{2}- y_{1}  }=\frac{z- z_{1} }{ z_{2}- z_{1}  }
     
  2. R!f@@

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    OMG !
     
  3. djsfantasi

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    Two or more, given the definition of an equation. Or you can write three equations. I see two equations if written in one way. If you add 3 variables, it can be written as five equations. By the way, per the definition of an expression, your example is not one.
     
  4. DerStrom8

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    An equation is a statement that one thing is equal to another. In the above example you are saying that (x-x1)/(x2-x1) is equal to (y-y1)/(y2-y1) and that (y-y1)/(y2-y1) is equal to (z-z1)/(z2-z1). There's two, and then you are also insinuating that (x-x1)/(x2-x1) is also equal to (z-z1)/(z2-z1). That's three.
     
  5. djsfantasi

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    But is the third equation necessary, as it can directly be derived from the first two?
     
  6. MrChips

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    We recognize this as the equations representing a straight line in 3-D space connecting two points, (x1,y1,z1) and (x2,y2,z2).

    These represent the orthogonal projections of the line on to the three planes,
    x = 0
    y = 0
    and
    z = 0

    Each projection in 2-D space is of the form:

    y = mx + c

    Hence there are three separate equations.
     
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  7. DerStrom8

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    It is insinuated. It may not be necessary to say because it's already proven, but it is an equation nonetheless.
     
  8. WBahn

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    Are they separate (i.e., independent)?

    If one equation is sufficient to define a line in 2D, why does it require two additional, independent equations to define a line in 3D?

    If you give me an equation that relates the y-coordinate to the x-coordinate and you give me another equation that relates the z-coordinate to the x-coordinate, what else is needed?

    Let's take the equations given in the OP and focus on the two obvious ones:

     \frac{x- x_{1} }{ x_{2}- x_{1}  } =\frac{y- y_{1} }{ y_{2}- y_{1}  }

     \frac{y- y_{1} }{ y_{2}- y_{1}  }=\frac{z- z_{1} }{ z_{2}- z_{1}  }

    It's obvious that I can combine these two equations trivially so as to get the equation

     \frac{x- x_{1} }{ x_{2}- x_{1}  } =\frac{z- z_{1} }{ z_{2}- z_{1}  }

    And therefore this is not an independent equation. Now, if by "separate" you mean something else, please specify what that is as I'm assuming you mean "independent".
     
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  9. WBahn

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    Since you aren't requiring that the equations be independent, it represents an infinite number of equations.
     
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  10. studiot

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    That's an interesting tongue-in-cheek answer.
    True of course, but one I never considered anyone would give.

    This is really a serious question, not a trick one and I am genuinely interested in people's various reactions and responses to it.
    Thus I have tried to be complete in my question, without asking in such a way as to prejudge the issue.

    So thank you everyone for your replies so far, keep them coming.
     
    Last edited: Apr 23, 2015
  11. DerStrom8

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    Once again, whether it is required or not, the question was asking how many equations are shown. The answer to that is three.
     
  12. wayneh

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    Only 2. The two equal signs provide information about the relationships between x, y and z.
    But you'd need another equation to hope to solve for the variables. And going from a=b and b=c to derive a=c, does not give you that 3rd equation.
     
  13. WBahn

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    Depends on what is meant by "solve". Instead of a single (i.e., a point) solution, the solution is an infinite set of points, called a line in this case, any of which satisfy all of the equations in the system.
     
  14. DerStrom8

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    Hmm, now that's a thought. Do the equal signs carry through? If not, you're right--since there's only 2 equal signs, there would be 2 equations. However, if you say x = y = z, does the first sign carry through since the second one is also an equal sign?

    Let me know if this doesn't make sense :p

    Matt
     
  15. MrChips

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    If I gave you only two equations of the form

    y = mx + c

    How would you determine the equation of the line in the third plane?

    There appears to me to be three independent equations.
     
  16. MrChips

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    Addendum:

    You can consider this to be one equation that defines the locii of all points falling on the line that intersects two points (x1,y1,z1) and (x2,y2,z2) in 3-D space.
     
  17. studiot

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    Yes it is the standard coordinate geometry expression to find the stright line between two points P(x1,y1,z1) and Q(x2,y2,z2) in 3-D space.

    The question I am exploring is

    Is it one equation or more and if more how many?
     
  18. wayneh

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    My point was only that ƒ(x)=ƒ(y)=ƒ(z) is NOT 3 independent equations, only 2. I arrived at that conclusion from knowing that, in practice, you cannot solve that for all 3 unknowns.

    Since one more fact (like x=5) would indeed allow solving for all 3 variables, there must already be two facts, ie. two equations.
     
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  19. WBahn

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    Here is what I think is the interesting point (and maybe the answer will come to me as I type).

    If I have

    Ax = By + C
    and
    Dx = Ez + F

    It is clear that I can change any of the six coefficients and, in doing so, just change one of the two equations. More specifically, I can alter the relationship imposed by one equation without altering the relationship imposed by the other.

    But if I have equations expressed in the form

    Gx = Hy + I = Jz + K

    Do I still have that ability, or does simply writing the equations in this manner limit what I can and can't do? My gut tells me the two ways of expressing the two relationships should be equivalent in all ways.

    I think I see how to show (one way or the other) which it is. I'll doo that in a bit.
     
  20. WBahn

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    I'll agree with you.
     
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