How many equations

WBahn

Joined Mar 31, 2012
29,979
I think I see how to show (one way or the other) which it is. I'll doo that in a bit.
Okay, the first step is to show that I can take the two decoupled equations

\(
Ax = By + C
\;
Dx = Ez + F
\)

This is trivial to do:

\(
\{Ax = By + C\}(D)
\;
\{Dx = Ez + F\}(A)
\)

\(
ADx = BDy + CD
\;
ADx = AEz + AF
\)

yielding

\(
ADx = BDy + CD = AEz + AF
\)

From this it's obvious that if I have the equations in coupled form, I can always decouple them, change one of them, and then recouple them as above. In doing so, I end up making changed to all three expressions (in general) but the changes to two of them cancel out so as to leave the relationship imposed by the corresponding equation unchanged.
 

WBahn

Joined Mar 31, 2012
29,979
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?
One of the two equations you give has to involve the third coordinate. Otherwise you can give a billion equations and not be able to say anything about the equation of the line in the third plane.

There appears to me to be three independent equations.
How can they be independent when, but definition, if one of them can be obtained via a combination of any of the others it is NOT independent?

I showed in Post #8 how you would take two of the equations and produce the third. Hence they are NOT independent no matter how much anyone might want them to be.

Now, let's take the equations for a straight line in two of the planes:

y = mx + b
z = ny + c

What is the equation of the line in the x-z plane?

z = n(mx+b) + c
z = (mn)x + nbc

Q.E.D.

These three equations are NOT independent. If you change just one of them, you do not obtain a different line, you now have a system of three independent equations whose solution is a point. If (in 3D) the solution is a line and you have three equations, then only two of them are independent.

EDIT: fix typos
 
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Thread Starter

studiot

Joined Nov 9, 2007
4,998
Thank you all for your responses, this is great.

The question arose because I said somewhere else that

There is not such thing as the equation of a (straight) line in 3 dimensions.

If we take the general form

ax + by = d

in 2 dimensions and extend it to 3 we get

ax + by + cz = d

This always defines a plane and a line is the intersection of two such planes, the coefficients of the line satisfying both planar equations.
This can only be obtained by solving a set of simultaneous equations as presented above. There are two independent equations to be solved.

This was as I learned it at university half a century ago.

Some modernists were quite vitriolic about this saying that the expression is now regarded as a single equation.
And indeed, looking round I see this view is being taught on at least some mathematics teaching websites.
 

WBahn

Joined Mar 31, 2012
29,979
Some modernists were quite vitriolic about this saying that the expression is now regarded as a single equation.
And indeed, looking round I see this view is being taught on at least some mathematics teaching websites.
I haven't heard this assertion, but I guess it wouldn't surprise me. I wonder if their thinking is along the lines of what I was musing about a few posts ago, but I think I satisfactorily addressed that issue in Post #20.

My office is in the midst of the Math department here, so I'll pose it to some of them and see what views they have. Most of them aren't in on Friday (we have a M/W, T/R class schedule for most classes), so it will probably be next week.
 

MrChips

Joined Oct 2, 2009
30,714
The solution for (x,y,z) is solved given a value of any one of the three variables x, y and z.
Now I see some semantics here.
Would we call this "one equation"
or "one set of equations" to solve for (x,y,z) given one parameter?
 

djsfantasi

Joined Apr 11, 2010
9,156
I say, two.

In the original problem statement as well in your simplified example, there are three expressions, which are in two equations.
 

MrChips

Joined Oct 2, 2009
30,714
I will have to conclude that in this and the OP there are two equations.

(x, y, z) are three unknowns.

A, B and C are three unknowns.

Hence we need three equations. The third equation arises when we supply a value for any one of the three unknowns.
 

DerStrom8

Joined Feb 20, 2011
2,390
To plumb the semantics on this a bit more. Let's say I start with

A=B
B=C

Is that two equations, or three?
It's definitely all in the semantics, but you all have changed my mind. I would have to agree that there are two. The third is insinuated, not written.

Matt
 

WBahn

Joined Mar 31, 2012
29,979
I will have to conclude that in this and the OP there are two equations.

(x, y, z) are three unknowns.

A, B and C are three unknowns.

Hence we need three equations. The third equation arises when we supply a value for any one of the three unknowns.
I agree that there are two equations.

The matter of how many equations we "need" is dependent on what we are trying to do. In this case there was no "what we are trying to do" specified, just the question of how many equations that set of symbols represents.

We often say that we "need" three equations to solve for three unknowns. But this is really a sloppy shorthand. First, we mean that we need three independent solutions, since having three equations that are not independent does achieve what we mean. But, furthermore, we mean that we need three independent to find a unique solution. The equation y=mx+b has a solution, that solution just happens to not be a unique pair of values <x,y>, but rather a set of points, that happen to form a unique line, in the x-y plane.
 

WBahn

Joined Mar 31, 2012
29,979
So I asked this of the Math Department Chair at lunch and he immediately said that it depends on what you mean by "an equation", but that for most reasonable interpretations the answer would be either two or three. I tried to ask about the school of thought that says it is just one equation but that was as we were leaving and he apparently didn't hear me.
 

djsfantasi

Joined Apr 11, 2010
9,156
So I asked this of the Math Department Chair at lunch and he immediately said that it depends on what you mean by "an equation", but that for most reasonable interpretations the answer would be either two or three. .
I like your Math Department Chair!

Two or more, given the definition of an equation.
By the way, this reminds me of an old Math Department joke. "How much is 2 + 2"? Inevitably, someone answers, "4." And the retort is, "5! ...for very large values of 2 and very small values of 5."
 

WBahn

Joined Mar 31, 2012
29,979
By the way, this reminds me of an old Math Department joke. "How much is 2 + 2"? Inevitably, someone answers, "4." And the retort is, "5! ...for very large values of 2 and very small values of 5."
The variant I always like was "Five, for sufficiently large values of 2."
 

WBahn

Joined Mar 31, 2012
29,979
Isn't that the same as, "I don't know"? o_O
Not really. It's acknowledging that the question is not sufficiently defined (bounded) so as to permit a singular answer that addresses the question that the person asking the question SHOULD have asked instead.

As I tell students all the time, the safest -- and often most accurate -- answer to most engineering questions is, "It depends."

It's not as hard to set the bounds on something as to provide a singular answer.
Sure, but then you risk answering a question that wasn't asked (happens around here -- and in real life -- all the time).
 
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BR-549

Joined Sep 22, 2013
4,928
I was taught that is one equation and that all terms are needed.

If you disregard one of the terms, the disregarded term could break equality and you wouldn't know it.

Improper tool use.
 

WBahn

Joined Mar 31, 2012
29,979
So, if

A=B=C

is one equation and

A=B
B=C

are two equations, then what is the difference between them? What information does one convey that the other does not?
 
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