Proportional Relationship between y, x, and z

In summary, when y is directly proportional to x at a constant z and y is also directly proportional to z at a constant x, then y is directly proportional to the product of x and z. This is because the two statements can be represented as y = kx and y = kz, where k is a constant. Multiplying these equations gives y = kxz, indicating that y is directly proportional to xz. However, if y^2 is directly proportional to xz, this would lead to the equation y = ky^2, regardless of the first two statements. Therefore, y^2 cannot be directly proportional to xz.
  • #1
santa
18
0
if [tex]y\propto x[/tex] at z constant

and [tex]y\propto z[/tex] at x constant

then

[tex]y\propto xz[/tex]



why not

[tex]y^2\propto xz[/tex]


thank you
 
Last edited:
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  • #2
if [tex] y^2 \propto xz [/tex]

then you would get

[tex] y \propto \sqrt{xz} [/tex]

so keeping fx. x constant you have

[tex] y \propto \sqrt{z} [/tex]

which is wrong. Maybe a proof could go like this:

assume:

[tex] y \propto y [/tex] for constant z

and

[tex] y \propto z [/tex] for constant x

this must meen that we can write

[tex] y(x,z) = f(z) x[/tex] for some function f and
[tex] y(x,z) = g(x) z[/tex] for some function g

then

[tex] g(x) x = f(x) x [/tex] so for x different from zero you have

[tex] g(x) = f(x) [/tex]

that is

[tex] y(x,z) = f(z) x[/tex]
[tex] y(x,z) = f(x) z[/tex]

so

[tex] y(x,1) = f(1) x[/tex]
[tex] y(x,1) = f(x) 1[/tex]

from which you get

[tex] f(x)= f(1) x[/tex], inserting this you have

[tex] y(x,z) = f(1) z x[/tex]

which is to say

[tex] y(x,z) \propto z x[/tex]

maybe the proof is flawed did it pretty sloppy.
 
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  • #3
santa said:
why not

[tex]y^2\propto xz[/tex]

Because then you'd have [itex]y\propto \sqrt{x}[/itex] for a fixed z. However, you can have [itex]y \propto x f(x)[/itex] where [itex]f(z)[/itex] is just-about-any function of z.
 
  • #4
thanks

but a have another


Definition of directly proportional - can k be negative?

In almost all textbooks, "directly proportional" is defined by saying
that a is directly proportional to b if and only if a = kb for some
constant k. That's perfectly sensible, but taking the definition
literally, it would seem to imply that any k will do, even negatives.

However, in every example that I have seen to illustrate the concept,
the term "directly proportional" is always applied to the relationship
between two positive quantities or two negative quantities--never
between a positive quantity and a negative quantity.
 
  • #5
Yes the constant of proportionality can take any value, positive, negative, real, complex.
 
  • #6
ok the constant of Hooke's_law

[tex]F=-KX[/tex]

k positive, negative, real, complex. or not
 
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  • #7
santa said:
ok the constant of Hooke's_law

[tex]F=-KX[/tex]

k positive, negative, real, complex. or not
Real and positive.
 
  • #8
santa said:
if [tex]y\propto x[/tex] at z constant

and [tex]y\propto z[/tex] at x constant

then

[tex]y\propto xz[/tex]



why not

[tex]y^2\propto xz[/tex]


thank you

Would that not lead to:-

[tex]y\propto y^2[/tex]

Irregardless of the first two statements.

Or have I grossly missed the point :smile:
 

1. What is a proportional relationship?

A proportional relationship is a mathematical relationship between two or more variables in which the ratio of one variable to another remains constant. In other words, as one variable increases or decreases, the other variable changes in proportion to it.

2. How do you determine if a relationship is proportional?

To determine if a relationship is proportional, you can create a ratio between the values of the two variables. If the ratio remains constant for all values, then the relationship is proportional. Another way is to graph the data and see if the points fall on a straight line passing through the origin.

3. What is the formula for a proportional relationship?

The formula for a proportional relationship is y = kx, where k is the constant of proportionality. This means that for any value of x, the corresponding value of y is equal to k times x.

4. Can a proportional relationship have a negative constant of proportionality?

Yes, a proportional relationship can have a negative constant of proportionality. This means that as one variable increases, the other variable decreases in proportion to it. For example, if y is always half of x, the constant of proportionality would be -0.5.

5. How can proportional relationships be represented graphically?

Proportional relationships can be represented graphically by plotting the values of the two variables on a coordinate plane and connecting the points with a straight line passing through the origin. The slope of the line represents the constant of proportionality, and the line will pass through the origin if the relationship is proportional.

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