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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.