What is the principle of algebra called?

[24forChromium]

I don't even know what is the general name for the thing whose specific name I am trying to find out, but in an algebra question, say 3x = 6y, there is the "assumption" that by dividing both side by 3, we will get x = 2y and this equation is as correct as the original one. What is thing that establishes the "shared correctness" of these equations? (Any vocabulary that I should have said but do not know would be helpful)

 

[micromass]

So what you are probably talking about is that both equations have the same solution set. This is a big thing in math, since we always want to preserve the solution set whenever possible. So if some values satisfy the first equation, then the exact same values should satisfy the second.
Sadly, not all operations preserve the solution set, and some of those operation are useful. For example
$$x=1~\Rightarrow~0\cdot x = 0\cdot 1$$
enlarges the solution set. While eliminating $x$ on both sides:
$$x^2 = x~\Rightarrow~x=1$$
makes the solution set smaller.

 

[Mark44]

I don't even know what is the general name for the thing whose specific name I am trying to find out, but in an algebra question, say 3x = 6y, there is the "assumption" that by dividing both side by 3, we will get x = 2y and this equation is as correct as the original one. What is thing that establishes the "shared correctness" of these equations? (Any vocabulary that I should have said but do not know would be helpful)

The property you're talking about is the multiplication property of equations. Multiplying both sides of an equation by the same nonzero constant yields a new equation with exactly the same solution set. Note that division by 3 is the same as multiplication by 1/3.

 

[24forChromium]

So what you are probably talking about is that both equations have the same solution set. This is a big thing in math, since we always want to preserve the solution set whenever possible. So if some values satisfy the first equation, then the exact same values should satisfy the second.
Sadly, not all operations preserve the solution set, and some of those operation are useful. For example
$$x=1~\Rightarrow~0\cdot x = 0\cdot 1$$
enlarges the solution set. While eliminating $x$ on both sides:
$$x^2 = x~\Rightarrow~x=1$$
makes the solution set smaller.

I am only starting to learn calculus, so I had honestly no idea what you are saying, I hope that do not make you feel offended or frustrated, my purpose is quite simply looking for words to use when I describe how the correctness of equation can be propagated for a paper that I am writing.

 

[24forChromium]

The property you're talking about is the multiplication property of equations. Multiplying both sides of an equation by the same nonzero constant yields a new equation with exactly the same solution set. Note that division by 3 is the same as multiplication by 1/3.

I found your answer somewhat helpful, "multiplication property of equations" sounds like something that I can use. Something similar must apply to addition too, and there may be a name that include both properties, maybe something like the "arithmetic property of equations" will do it. Something else I am interested in is the fact that "correctness" or as you call it, "solution sets" are shared across equations that can "become" one another. In the example of x = 2y, the truth of this equation is shared with the truth of 3x = 6y, but if x and y are actually physical variables, such as velocity and air friction, by disproving one of the two equations, the other is also disproved, what justification can i provide for the falsification/verification of the other equation?

 

[Mark44]

I found your answer somewhat helpful, "multiplication property of equations" sounds like something that I can use. Something similar must apply to addition too, and there may be a name that include both properties, maybe something like the "arithmetic property of equations" will do it.

The two are separate, I'm pretty sure: addition property of equations and multiplication property of equations. For the addition property, you can add negative numbers, so a separate property for subtraction isn't needed. There are similar properties for inequalities, although multiplying by a negative number changes the direction of the inequality.

Something else I am interested in is the fact that "correctness" or as you call it, "solution sets" are shared across equations that can "become" one another.

The right term is "equivalent equations" -- equations that have identical solution sets. The equations x = 2, x + 3 = 5, and 2x - 1 = 3 are all equivalent equations -- same solution set; namely {2}. I can transform any of these equations into any of the others by applying one or more of the properties I mentioned.

In the example of x = 2y, the truth of this equation is shared with the truth of 3x = 6y, but if x and y are actually physical variables, such as velocity and air friction, by disproving one of the two equations, the other is also disproved, what justification can i provide for the falsification/verification of the other equation?

It doesn't matter whether the variables represent physical quantities. The only important thing is the underlying mathematics. The equations x = 2y and 3x = 6y are equivalent, so for any pair of numbers (x, y) that is a solution of the first equation, the same pair will be a solution of the second equation. Conversely, if a pair (x, y) is NOT a solution of one equation, it will also not be a solution of the second.

The equations of your examples are very simple: their graphs represent straight lines in the plane. Pairs of numbers that are solutions are points on the graph of the line. Pairs of numbers that aren't solutions are not on the graph of the line.

 

[mathwonk]

this property is called the cancellation property in algebra, i.e. that if AX=AY and A≠0, then X=Y. It holds when the multiplier is not a zero divisor, or for any non zero multiplier in an "integral domain". I.e. a ring of numbers is called an integral domain if whenever A≠0 and Ax=Ay, then x=y.

 

[symbolipoint]

I am only starting to learn calculus, so I had honestly no idea what you are saying, I hope that do not make you feel offended or frustrated, my purpose is quite simply looking for words to use when I describe how the correctness of equation can be propagated for a paper that I am writing.

"Only starting to learn calculus" means you understand the multiplication property of equality and the other properties which are learned in beginning basic and intermediate algebra.
... otherwise, what happened? How did you get into Calculus 1? What pathway of courses lead you to Calculus 1?

 

[nuuskur]

This is something along the lines of Gauss's statements about solving system of equations. Right now we have a system of one equation:
$6y - 3x = 0$
If we conduct elementary row operations on the system's matrix, then the resulting matrix is Equivalent to the original one.
As we only have one row to work with, we can multiply the row with any non zero scalar:
$k6y - k3x = k0 = 0 , k\neq 0$

 

[jbriggs444]

I don't even know what is the general name for the thing whose specific name I am trying to find out, but in an algebra question, say 3x = 6y, there is the "assumption" that by dividing both side by 3, we will get x = 2y and this equation is as correct as the original one. What is thing that establishes the "shared correctness" of these equations? (Any vocabulary that I should have said but do not know would be helpful)

Going off on a different tangent that may capture what you are after...

The idea that dividing both sides by a constant gives a new equality that is also valid can be derived from the rules of "first order logic with equality" (https://en.wikipedia.org/wiki/First-order_logic#Defining_equality_within_a_theory)

Starting with $3x = 6y$ you could assert (by the reflexive property) that $\frac{3x}{3} = \frac{3x}{3}$. Then you could use substitution for formulas to replace the "$3x$" on the right hand side with "$6y$". That gives you the desired result: $\frac{3x}{3} = \frac{6y}{3}$.

 

[Ssnow]

If I remember '' equivalence principles '' for an equation or something like that ...