Is the following equation regarded as a linear equation?


The original question is as below:

Solve the linear system given by the following augmented matrix:

$\left(\begin{array}{ccc|c}2 & 2 & 3 & 1 \\2 & 5 & 3 & 0 \\0 & 0 & 0 & 5\end{array}\right)$

Note the words linear system in the original question. So, I was asking myself whether $0x_1+0x_2+0x_3=5$ is a linear equation. Can we call all of the equations given by the matrix collectively as a linear system?

  • 2
    $\begingroup$ I would say that it is an unsolvable linear equation (yet a linear equation). $\endgroup$ – Carlos Eugenio Thompson Pinzón Dec 3 '13 at 14:18
  • $\begingroup$ I should not call that an equation. A countertruth could be a better name ! $\endgroup$ – Claude Leibovici Dec 3 '13 at 14:22
  • $\begingroup$ $0=5$ is a linear equation? how about $0x^2=5$? Both of them can be written as $0=5$? $\endgroup$ – shuxue Dec 3 '13 at 14:39

The reason you MUST call it a linear equation is that you want to call the following a linear equation, for all constants $a_1, a_2, a_3$: $$a_1x_1+a_2x_2+a_3x_3=5$$ You don't want to call this a "sometimes" linear equation.

  • $\begingroup$ However, some books tend to write $0x_1+0x_2+0x_3=5$ as $0=5$. How about $0x^2=5$? Both of them can be written as $0=5$? $\endgroup$ – shuxue Dec 3 '13 at 14:46
  • $\begingroup$ $0=5$ may be a linear equation, depending on context. $0x_1=5$ is definitely a linear equation, and that's the type you have here. $\endgroup$ – vadim123 Dec 3 '13 at 14:48
  • $\begingroup$ The book 'Elementary linear algebra, Applications Version' 10th edition, by Anton defined a linear equation in the variables $x_1, x_2, ..., x_n$ as an equation of the form $a_1x_1+a_2x_2+...+a_nx_n=b$, where $a_1, a_2, ..., a_n$ are coefficients of the linear equation and these coefficients are not all zero. $\endgroup$ – shuxue Dec 4 '13 at 15:18
  • $\begingroup$ What about $$0x_1 + 0x_2 + 0x_3 = 0$$ Is this also a linear equation? Every point of 3-space is a solution so this expression fills all 3 space. $\endgroup$ – neofoxmulder Feb 15 '14 at 9:32

It is a linear equation with no solutions, so the linear system has no solutions.


I would rather see $x$, $y$ and $z$ instead of $x_1$, $x_2$ and $x_3$... But it is still a linear equation because the degree of the polynomial is $1$. I know what you are stuck on, the fact that the coefficients are $0$, therefore making the left-hand-side evaluate to $0$, which is degree $0$. But because of the variables, we have to call it a linear equation. Just think of it as a linear equation with no solutions (because $0$ $\neq$ $5$).


Quadratic equations are of the form $Ax^2 + Bx + C = 0$, where $A$, $B$ and $C$ are all real numbers. The thing different about this equation is that the degree is $2$, not $1$. Similarly, cubic equations have degree $3$, quartic equations have degree $4$, and so on. Equations of degree $0$ have no special name.


Many people get confused when they see things like $x_1$, $x_2$, and $x_3$. Whenever you see something like that, think of it as just being $x$, $y$ and $z$. They are different variables and are in no way related to each other.


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