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Section 1-1 : The 3-D Coordinate System

4. List all of the coordinates systems (\(\mathbb{R}\), \({\mathbb{R}^2}\), \({\mathbb{R}^3}\)) that the following equation will have a graph in. Do not sketch the graph.

\[7{x^2} - 9{y^3} = 3x + 1\] Show Solution

First notice that because there are two variables in this equation it cannot have a graph in \(\mathbb{R}\) since equations in that coordinate system can only have a single variable.

There are two variables in the equation so we know that it will have a graph in \({\mathbb{R}^2}\).

Likewise, the fact that the equation has two variables means that it will also have a graph in \({\mathbb{R}^3}\). Although in this case the third variable, \(z\), can have any value.