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Consider the equation $x+3=3(x+1)-2x$. Is this equation a linear equation?

I would say to my students that an equation in 1 variable is linear when it can be simplified to the form $ax=b$ where $a\neq0$. Also, I would mention that a linear equation in 1 variable has only 1 solution. Therefore, I would say that the equation $x+3=3(x+1)-2x$ is not linear because it can be reduced to $0x=0$ which has infinite solutions.

However, a colleague whose research area is algebra says that a linear equation is one that only involves polynomials of degree 1. Therefore, the equation $x+3=3(x+1)-2x$ is linear. Then, according to the definition of my colleague, the equation $x-x=0$ is linear as well.

I was thinking that maybe a third possibility is that according to the definition of "linear equation" only equations of the form "expression$=0$" can be classified as linear or non-linear. In this case, the equation $x+3=3(x+1)-2x$ is not linear nor non-linear, but the equation $x-x=0$ would be linear.

What should be the definition for "linear equation" in 1 variable suitable for a math course?

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    $\begingroup$ Does it matter? What do you need a precise definition of "linear equation" for, anyways? $\endgroup$
    – Eric Wofsey
    Feb 1, 2021 at 23:52
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    $\begingroup$ As far as I am concerned, you are right and your colleague is wrong. I do not consider $x + (-x) = 0$ to be a linear equation. Unclear if others will agree. Also unclear if there is only 1 answer, or whether the answer depends on a (non-universal) convention. $\endgroup$
    – user2661923
    Feb 1, 2021 at 23:52
  • $\begingroup$ Such a definition is usually just a tool for classifying equations. The courses I’ve come across would state the most general formula for an equation (for example, $y’’+py’+qy=0$ Is the most general 2nd-Order linear ODE). So I guess the $ax=b$ idea is not bad. because it corresponds to standard definitions in linear algebra. Of course, never test the students on these definitions... only test them on problem solving ability $\endgroup$
    – Benjamin Wang
    Feb 1, 2021 at 23:53
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    $\begingroup$ Would you rename the field to "Linear Algebra as well as the Algebra of Polynomial Equations Having Degree Zero"? $\endgroup$
    – Théophile
    Feb 2, 2021 at 0:00
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    $\begingroup$ I would definitely call the equation $x+3=3(x+1)-2x$ linear. A "linear system of equations" is a system of equations of the form $Ax = b$, where $A$ is an $m \times n$ matrix and $b$ is an $m \times 1$ column vector. There is no requirement for $A$ or $b$ to be nonzero. One might alternatively define a "linear equation" to be an equation of the form $L(x) = b$, where $L$ is a linear transformation. And there is no requirement for $L$ to be nonzero. $\endgroup$
    – littleO
    Feb 2, 2021 at 1:36

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However, a colleague whose research area is algebra says that a linear equation is one that only involves polynomials of degree 1. Therefore, the equation $0=0$(on simplification)  is linear. Then, according to the definition of my colleague, the equation  is linear as well.

The problem is "The degree of a zero polynomial is not defined or $-\infty$". So saying that zero polynomial has degree $1$ is little vague as $0\equiv 0x\equiv 0x^2\equiv 0x^3\equiv.....$

What should be the definition for "linear equation" in 1 variable suitable for a math course?

A linear polynomial $p(x)=ax+b, a\ne 0$ when equated to a real number $\alpha(say)$ i.e. $p(x)=\alpha$ is called a linear equation.

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