# Teaching linear algebra. Is this equation a "linear equation"?

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?

• Does it matter? What do you need a precise definition of "linear equation" for, anyways? Feb 1, 2021 at 23:52
• 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. Feb 1, 2021 at 23:52
• 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 Feb 1, 2021 at 23:53
• Would you rename the field to "Linear Algebra as well as the Algebra of Polynomial Equations Having Degree Zero"? Feb 2, 2021 at 0:00
• 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. Feb 2, 2021 at 1:36

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 $$0\equiv 0x\equiv 0x^2\equiv 0x^3\equiv.....$$
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.