Is $0x_1+0x_2+0x_3=5$ a linear equation?

Question

Is the following equation regarded as a linear equation?

$$0x_1+0x_2+0x_3=5$$

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?

Answer 1

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.

Answer 2

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

Answer 3

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 $0$ $\neq$ $5$).

EXTRA INFORMATION

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.

Tip:

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.