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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.

Parthiv Basu
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Andrew
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  • In three dimensions, the graphs of $x=0$ And $x+2y=0$ are also planes. It doesn’t make sense to speak of the number of “dimensions” of the solution without first specifying the ambient space. – amd Aug 03 '19 at 01:12

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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.

Vamshi Kumar Kurva
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