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I am trying to find the correct math term to describe the relationship between equations that have the same “form.” I used the word form but was told that it was incorrect. Let me give you an example of what I refer to as the same form. There are 3 equations:

EQ1: X = a + b + c+ zY.

EQ2: X = a + b + c + d + e + wY.

EQ3: X = a + b + c+ + d + e + f + g + uY.

Each equation can be said to have 3 sections: Section A [X], Section B [series of summation of values] and Section C [value times variable Y]. I can show that a characteristic of one of the equations [A - C does not equal B, since B does not contain variables X, Y] is representative of all the equations since they have the same form: Sections A, B and C.

The examples are just for discussion. They are not meant to be actual math problems.

What is the correct math term for this sentence: “All the equations have “the same” / ”a similar” [blank], so the results shown for equation 1 represent the result for all of the equations.”

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    $\begingroup$ I'd say they have the same form: in this case, they're all linear, with three different types of terms: those involving X, those involving Y (include a coefficient), and the constant (with neither X nor Y) and those involving Y $\endgroup$
    – Jules Lamers
    Commented Oct 11 at 8:45
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    $\begingroup$ "Form" is the correct word. It is likely that something about your explanation of the form was not precise or was otherwise incorrect. If possible, you should ask the person who gave you the feedback how you can improve your statement, or show us exactly what you said that caused you to get the feedback. Good luck! $\endgroup$
    – Chris Cunningham
    Commented Oct 11 at 15:19
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    $\begingroup$ Form, kind, type, etc. It is all the same. There is no particular term that is better or worse than any other one as long as it delivers the message in an unambiguous way and is not too far from the usage of the same word in common English. Don't overthink it! $\endgroup$
    – fedja
    Commented Oct 11 at 15:48
  • $\begingroup$ I appreciate the comments however I have been told directly that "form" is wrong to describe these equations. I am trying to find the "math" term. Any suggestions for a math term? $\endgroup$
    – Bob Smith
    Commented Oct 14 at 7:36
  • $\begingroup$ We often talk about "identification" when we have a canonical representation which is unique. For instance, two complex numbers $a+ib$ and $c+id$ with $a,b,c,d$ real are equal iff $a=c$ and $b=d$; we say that we can "identify" the real and imaginary parts. Two polynomials $P = \sum a_k X^k$ and $Q = \sum b_k X^k$ are equal iff $\forall k,\, a_k=b_k$; we say that we can "identify" the coefficients. Similarly, if you have proven that two equations given in some standard form have the same solutions iff they have the same coeffs, then you can talk about identification of the equations. $\endgroup$
    – Stef
    Commented Oct 14 at 9:14

2 Answers 2

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I used the word form but was told that it was incorrect.

This usage is not incorrect in the sense that everyone uses it and understands what it means. For example:

  • Anton & Rorres (Linear Algebra): Equation 13 has the same form as Equation 12.
  • Gallian (Abstract Algebra): Since the sum of two polynomials of the form ax²+bx+c is another one of the same form.
  • Apostol (Calculus): Notice that this has exactly the same form as (5.18).
  • Moise (Elementary geometry): Comparing the formulas x' = ax - by, y' = bx + ay ... we see that these have the same form x = a' x' - b'y', y = a'y' + b'x, where a'=a and b'=-b.
  • Suppes (Logic): the line corresponding to line (n+1) ... is going to have the same form in every indirect proof.
  • Boyce & DiPrima (ODE): Eq. (7) has the same form as Eq. (1).
  • Evans (PDE): the boundary-value problem (1), (2) converts into a problem having the same form.

Well, I'll stop here believing that we are justified in making any use that has the same form.

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So If I am understanding your question correctly,'form ' is not a mathematical term and you want some word to talk about all these similiar equations and from your last paragraph you need to do that in order to show they follow similar result. Although if the context is clear using 'form' wouldn't be incorrect

In this particular case, you are writing equations of the form $X =\alpha + \beta Y $ as all the $a$'s + $b$'s will be a constant, so you can actually just write: $$\begin{aligned} X &= (a + b + c) + dY \\ X &= (a + b + c+ + d + e) + fY \\ X &= (a + b + c+ + d + e + f + g) + uY \end{aligned}$$

are of the form $X =\alpha + \beta Y $ ...and hence they follow similar result, or whatever

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