But is the third equation necessary, as it can directly be derived from the first two?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.
It is insinuated. It may not be necessary to say because it's already proven, but it is an equation nonetheless.But is the third equation necessary, as it can directly be derived from the first two?
Are they separate (i.e., independent)?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.
Since you aren't requiring that the equations be independent, it represents an infinite number of equations.How many equations would you say this expression represents 1,2, 3 or more?
\( \frac{x- x_{1} }{ x_{2}- x_{1} } =\frac{y- y_{1} }{ y_{2}- y_{1} }=\frac{z- z_{1} }{ z_{2}- z_{1} }\)
That's an interesting tongue-in-cheek answer.Since you aren't requiring that the equations be independent, it represents an infinite number of equations.
Once again, whether it is required or not, the question was asking how many equations are shown. The answer to that is three.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?
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.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.
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?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.
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.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.
Here is what I think is the interesting point (and maybe the answer will come to me as I type).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?
I'll agree with you.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.
by Jake Hertz
by Aaron Carman
by Jake Hertz