A system of m equations in n unknowns, can all the coefficients of one unknown be zero?

Question

Let's say $m = 2, n = 3$:

$x + y + 0z = 1 $

$x + y + 0z = 2$

Is this still considered to be a system with $3$ unknowns? Or does this collapse to $2$ unknowns?

Answer 1

This is a matter of convention. In some contexts zero is treated as nothing and in other contexts it is just another number. For example, The polynomial $\;ax^2+bx+c\;$ is a quadratic in general, but what if $\;a=0?\;$ In one context, the polynomial is regarded as $\;bx+c\;$ and is clearly linear, and in another context, it is an element of the linear space of polynomials with degree at most two.

In the context of two unknowns, $\;x\;$ and $\;y,\;$ the equation $\;x+0y=x=1\;$ has a unique solution for $\;x\;$ and the unknown $\;y\;$ is indeterminate. If there are any other unknowns, then their values are still indeterminate. In general, some uknown variables can be uniquely determined by linear equations, and others have indeterminate values and still unknown. In contrast, the system of equations $\;x=1,\;x=0\;$ is inconsistent and the value of $\;x\;$ is not unknown, but undefined,

Answer 2

You are right. This provided example just collapses to a system of 2 equations within 2 unknowns. The third there mentioned unknown $z$ is not concerned by either of these equations, and so is left fully free. But @HagenvonEitzen is right too, nothing is said by the full and the reduced system about any further, not even being mentioned variables.