How many equations

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

1. studiot Thread Starter AAC Fanatic!

Nov 9, 2007
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How many equations would you say this expression represents 1,2, 3 or more?

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

Apr 2, 2009
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OMG !

3. djsfantasi AAC Fanatic!

Apr 11, 2010
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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 Well-Known Member

Feb 20, 2011
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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 AAC Fanatic!

Apr 11, 2010
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But is the third equation necessary, as it can directly be derived from the first two?

6. MrChips Moderator

Oct 2, 2009
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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 Well-Known Member

Feb 20, 2011
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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 Moderator

Mar 31, 2012
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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 Moderator

Mar 31, 2012
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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 Thread Starter AAC Fanatic!

Nov 9, 2007
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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 Well-Known Member

Feb 20, 2011
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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 Expert

Sep 9, 2010
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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 Moderator

Mar 31, 2012
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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 Well-Known Member

Feb 20, 2011
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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

Matt

15. MrChips Moderator

Oct 2, 2009
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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 Moderator

Oct 2, 2009
12,648
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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 Thread Starter AAC Fanatic!

Nov 9, 2007
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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 Expert

Sep 9, 2010
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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 Moderator

Mar 31, 2012
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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 Moderator

Mar 31, 2012
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I'll agree with you.