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I am more of an Intuitive Learner and Teacher so while looking at Linear Equations chapter I see that they are teaching Equations of Line there, which in my opinion is wrong.

See How I understand it is:

  • Equation of Line

y = mx + c : This in a way gives you the address of the Street(Line). WHERE this line lies in XY Plane.

Whereas

  • Linear Equation

ax + by + c = 0 : This gives a way to verify whether a House(Point) lies in the Aforementioned Street.

Another difference I see is, Equation of a Line is relation between VALUES on x and y axis, whereas Linear Equation deals with POSITIONS(projections of x-y values on XY plane) on the XY planes. In a way Linear Equation represents the solution set for Equation of a Line.

Am I correct in my understanding? Are Linear Equations and Equations of a Line, one and the same thing? Any sources where I can get more clarification on these.

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    $\begingroup$ Much more important is that $y = mx + c$ gives equations of non-vertical lines. Moreover, "equation of a line" is when a geometric connection via a coordinate plane is involved, whereas "linear equation" is a purely algebraic notion in which no coordinate plane is necessarily in the background. Finally, if you want the solution set, then there already is standard notation students that would have seen. I think the distinctions you're trying to make are rather artificial and based more on certain limited uses (continued) $\endgroup$
    – Dave L Renfro
    Commented Jun 14, 2022 at 6:24
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    $\begingroup$ you have in mind than on matters of mathematical significance (i.e. for "equation of a line" you seem to want to plug numbers on the right hand side to obtain a value for the left hand side, whereas for "linear equation" you seem to want to plug numbers on both sides and determine whether equality is true -- a process one can also do for "equation of a line"). $\endgroup$
    – Dave L Renfro
    Commented Jun 14, 2022 at 6:28
  • $\begingroup$ Notice that $$x=y=z,$$ which represents a line in 3D space, comprises two linear equations of the form $ax+by+cz+d=0.$ $\endgroup$
    – ryang
    Commented Jun 14, 2022 at 10:57
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    $\begingroup$ From a mathematical point of view, an "equation of a line" is simply any equation whose solution set is a line. Meanwhile, a linear equation is any equation where every summand involving a variable is linear. The "usual" equation of a line $y=mx+c$ very much is a linear equation, so it's more than fair to include it in a chapter about linear equations. And the equation $ax+by=c$ has a line as its solution set (provided a and b aren't both 0) so including it in a chapter about lines is also perfectly fine. $\endgroup$
    – Vercassivelaunos
    Commented Jun 14, 2022 at 12:29
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    $\begingroup$ General answer for teaching elementary math.... Get a good textbook and follow it exactly. $\endgroup$
    – Gerald Edgar
    Commented Jun 15, 2022 at 0:53

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The only "real" difference (except for maybe some analogies that work differently as in your house/street example) is that the linear equations are a proper superset of the equations of lines because you can't describe vertical lines with $y=mx+c$, but you get them with $ax+by+c=0$ for $b=0$.

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    $\begingroup$ But these are all linear equations. There are only two kinds of linear equations (of two variables) which are not equations of lines: The always true equations of the form $a=a$ describe the whole plane, while the always false equations of the form $a=b$ with $a\neq b$ describe the empty set. All other linear equations of two variables have lines as their solution set. $\endgroup$
    – Vercassivelaunos
    Commented Jun 14, 2022 at 19:00
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I suspect that two issues may be partly underlying your intuition:

  • First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $x$ and $y$. This is a meaningful distinction, one worth making. However, the correct words for making this distinction are not "equation of a line" vs. "linear equation" (which mean exactly the same thing) but rather "function" vs. "relation". As other commenters have pointed out, the format $y = mx + c$ only describes non-vertical lines (which are functions), whereas $ax + by + c = 0$ also includes vertical lines (which are not functions). Another way of describing this distinction is that $y=mx + c$ defines $y$ explicitly in terms of $x$, whereas $ax + by + c = 0$ defines $y$ implicitly in terms of $x$ (assuming $b \ne 0$).
  • Second, in the equation $y = mx + c$ the parameters $m$ and $c$ are uniquely determined by the non-vertical line. That is: given any non-vertical line, there is a unique pair of numbers $(m, c)$ with the property that $y = mx + c$ describes the line; and conversely, given any pair of numbers $(m, c)$ there is a unique line given by the equation $y = mx + c$. In marked contrast, the parameters $a, b, c$ in the equation $ax + by + c = 0$ are not uniquely determined; if $ax + by + c = 0$ is an equation for some line $\ell$, then for any non-zero value $k$, the equation $kax + kby + kc = 0$ describes the same line. I suspect this may be what your intuition is recognizing when you say that $y = mx + b$ is "the address" of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.

Having said all of this, it is definitely the case that the textbook is not wrong. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.

Finally, I want to commend you for asking this question. Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher. The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?

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  • $\begingroup$ Thanks for understanding and explaining... $\endgroup$
    – Ashish Shukla
    Commented Apr 27, 2023 at 10:16
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Your intuition is fundamentally not correct. These are all equations of a line, and your textbook is correct to include them in the same chapter. The term "Linear Equation" is used because it is, in fact, the Equation of a Line. It's a fact I take the time to prove in my college algebra courses: the graph of any linear equation in two variables is a line.

That said, there are different formats in which one can write a line, with various advantages and disadvantages in different use-cases; but they're all effectively equivalent, as can be shown with just a few lines of algebraic manipulation. Customary terminology includes the following:

  • Standard Form: $ax + by = c$. Pros: Used as the definition of a linear equation. Can represent any line, including vertical ones. Can be written without use of fractions for the coefficients. Expands nicely to other shapes, such as conic sections. Easy to find both x- and y-intercepts.

  • Slope-Intercept Form: $y = mx + b$. Pros: Has a clear graphical interpretation via use of slope. As a function, easy to compute dependent variable $y$ from known independent variable $x$. Can be easily estimated visually from a graph. Bootstraps to understanding derivative in calculus. Can be implemented directly in a computer programming language.

  • Point-Slope Form: $y - y_1 = m(x - x_1)$. Pros: Appears naturally midway into proof that nonvertical lines can be written in slope-intercept form. Shortcuts finding linear equation through two given points. Useful for other purposes in calculus.

There are other forms as well, and it would benefit you to know this standard terminology (shown in your textbook and elsewhere).

Related: Are there any good exercises on point-slope form of a linear equation by itself?

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They aren't very much different from some perspectives. If what you care about is the question "Is this point on the line or not?" then you can simply plug your $(x,y)$ values into either of them. If you get an equality, then yes. Otherwise, no. Here, the difference is that the $y=mx+b$ form cannot handle vertical lines while $Ax+By+C=0$ can. We often call $Ax+By+C=0$ the standard form because it can handle all lines and any two different triples $(A,B,C)$ will give different lines. For example, $y=mx+b$ corresponds to $(A,B,C)=(-m,1,-b)$. (You subtract $mx+b$ from both sides.)

However, if we can write a line in the form $y=mx+b$, then we have written $y$ as an explicit function of $x$. That is, you can pop an $x$ value in, do a few bits of arithmetic and get the corresponding value of $y$. The point is that you don't have to plug in $x$ then solve for $y$. If you started with $Ax+By+C=0$, you would. (It's not hard to do so, but that isn't the point.)

(Note: I use explicit as a contrast to the standard use of implicit. e.g. $x^2+y^2=1$ defines $y$ as an implicit function of $x$, away from $x=\pm1$. )

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Etymology/Grammar

"Line" is the root word, a noun whereas "linear" is a derived word, an adjective. So, apart from the grammar distinction, one could call "line" and "linear" the same word.

Life however does not always respect grammar nor maintain etymological connections ... including...

Math

In math the provenance of "line" is just well a line – a geometrical straight line.

"Linear" starts off there ­– lines and their equations in $\mathbb{R}^2$ and $\mathbb{R}^3$ – and then broadens considerably:

  • linear combinations in arbitrary vector spaces where the notion of (geometric) line may be quite far-fetched, eg $\mathbb{R}^n$, polynomials over a variable, etc
  • linear differential equations
  • linear difference equations
  • and so on.

In all such cases the connotations of "linear" are likely to be far removed from anything reasonably geometrical.

And so a more mathematically sophisticated rendering of your question would be to ask:

"Linear" in the context of which vector space?

[Aside: Though I suspect your students are not ready for this level of abstraction, it may still be useful for your own understanding]

The underlying issue I feel you're groping for is...

Normal Form

In basic school arithmetic we say "2 + 3 is 5" and write it as $2 + 3 = 5$. But most typically you will replace $2+3$ by $5$ and not vice versa.

So in some loose informal sense 5 is better than 2+3.

This informal notion is formalized in systems like lambda calculus, and rewriting systems where two sets are defined, a larger set of terms and a smaller subset of normal-form terms, and computation in the formal system proceeds until there are no rewriteable terms and only a normal form (also called ground term) remains.

[Note: "Ground term" is a downward metaphor — things fall downward till they hit the ground. The point they hit the ground is the ground term or normal form]

So in the lambda calculus $2+3$ would be rewritten to $5$ and $2 + 3 \rightarrow 5$ would be considered as a delta rewrite rule.

Notice that there is a strange double speak going on here:

  • Denotationally :: $ 2+ 3 = 5$ and $5 = 2+3$ are identical.
  • Computationally :: We prefer $2+3 = 5$ over $5 = 2+3$

If we want to be more rigorous we would write it with a $\rightarrow$ rather than a $=$.
So $2+3 \rightarrow 5$ is a rule in the formal rewrite system but $5 \rightarrow 2+3$ is not.

Typically mathematicians are not that rigorous, sometimes treating $2+3$ and $5$ as completely equivalent, and sometimes preferencing one over the other.

As soon as we go beyond simple numbers trouble starts. Yeah, we call $x^2 -5x + 6$ the simplified form and $(x-2)(x-3)$ the factorized form. But there's nothing intrinsically preferencing one over the other.

This gets only more knotty the more advanced we get.

And even returning to plain numbers: So we may naturally prefer $5$ over $2+3$ but between 12(base10) 14(base 8) and 1110(base2) there is little 'more normal' about the one than the other aside from convention.

Coming to your question:

Equations of a line

As Daniel Collins points out there are a bunch of different equation forms for straight lines

  • They are (generally) convertible between each other
  • each with their pros and cons

In short there are different forms of equation of a line. And no normal form.

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