Linear equation ax+b=c: why 3 constraints? why not 2 or 4?

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Asam

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1679819539442.pngHello, everyone
why in general form of linear equation we have 3 constants. Why not 2 or 4?

Thank you!
 
Asam said:
View attachment 35346Hello, everyone
why in general form of linear equation we have 3 constants. Why not 2 or 4?

Thank you!
Click to expand...
How would you represent such an equation with 2 or 4 constants? Show us, and we will have more to discuss.

My standard answer when someone asks "Why can't we do it this other way?" is "Try it and see. You may find why; or you may find it doesn't matter."
 
Where did you find 3 constants? I thought a general form is [imath]ax+d=0[/imath], which you would get by using [imath]d=b-c[/imath].
 
lookagain said:
Asam, please check your source. It should be \(\displaystyle \ ax + by = c, \ \) as in I expected you copied it incorrectly.
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ax +by = c is the general form of a linear function.

ax + b is the general form of a linear expression.

ax + b = c, eg 3x + 5 = 11, is the general form of a linear equation, which of course can be quickly changed into 3x - 6 = 0 (which still has 3 constants ie 3, -6 and 0).
blamocur said:
Where did you find 3 constants? I thought a general form is [imath]ax+d=0[/imath], which you would get by using [imath]d=b-c[/imath].
Click to expand...
Still has 3 constants, namely a, d and 0.
 
I learned (albeit decades ago) that all general forms have zero on one side, whereas standard forms have the variable terms or the constant separated. Since then, I've seen the names 'General' and 'Standard' defined in so many different ways that I try to accept whatever name the other person uses. Times have changed. Either of the following two forms might be called 'general' or 'standard'.

Ax + By + C = 0

Ax + By = C

I regularly see each of the following called 'standard', too.

y = mx + b

y - y1 = m(x - x1)

When I first read the op, I'd wondered why they had y=c, heh. :)

linEqns.PNG
[imath]\;[/imath]
 
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The main question here is, what does the OP mean by "general form", and (perhaps even more important) by "linear equation"? That's why I started with a question, and really can't say any more until we get that answer. My initial assumption would be that the form shown is what was taught, and therefore it determines what they must mean by "linear equation".

If he means a linear equation in one variable, then none of what has been said about equations of lines is relevant. But also, there are a couple different forms that could be called "general" (which happen to have 2 and 4 constants, interestingly enough), and if the form shown is what he was taught, then the ultimate answer is, "That's what some author chose to call the general form, but it is not the only choice".

In my experience, one of the least standardized words in mathematics is "standard"; and one of the least generally agreed upon words is "general". All definitions of these words are ultimately arbitrary, and I am not surprised if someone uses them differently than I might have learned, or might prefer if I were writing a book.
 
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