Relationship between the number of variables in the equation and the number of dimensions of its graph

Question

The graph of the equation $x = 0$ has one dimension, as it is a vertical line. The graph of the equation $x + 2y = 0$ has 2 dimensions, because it is just a negatively sloped line. The equation $x + 2y + 3z = 0$ is proven to be a plane (3 dimensions) through the utilization of dot product. Does this pattern continue with an increasing number of variables? Proofs or counterexamples would be appreciated.

Answer 1

If there is an equation in $n$ variables, then there are $n-1$ degrees of freedom, i.e choose any $n-1$ variables independently from the domain, then the last variable is a function of these $n-1$ variables. Hence the graph is in $n$ dimensions.