A line or a plane or a point?

Example 1: $y=3x-2$

Is the graph of $y=3x-2$ the graph of a line or a plane or a point? You might recognize this as a familiar form of the equation of a line with slope 3 and $y$-intercept $-2$, as in the following graph.

However, if we think $y=3x-2$ is a line, we have implicitly assumed that the equation $y=3x-2$ is an equation in two dimensions, i.e., in the two variables $x$ and $y$. It could actually be an equation in three dimensions, say with the variables $x$, $y$, and $z$. Although the $z$ doesn't appear in the equation, we could think of the equation as having a $z$ with a coefficient of zero. We could write the equation as $y=3x+0z-2$. Here's what plotting the equation $y=3x-2$ in a three-dimensional coordinate system looks like.

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An angled line or a plane. The graph of the equation $y=3x-2$ looks like a line, which it would be if were an equation in two dimensions, i.e., in the $xy$-plane. However, rotate the graph with the mouse to give you a new perspective on the graph. Since in this case, the graph is really in three dimensions, the graph of the equation becomes a plane. The equation just happens to not depend on the variable $z$, so the plane is parallel to the $z$-axis. You can press the Home to key to return to the original view.

Before rotating, the graph looks like the previous one, as the $z$ dimension is hidden. The fact that the graph is a plane doesn't become obvious until you rotate the axis.

We conclude that we don't know what the graph of $y=3x-2$ is unless we specify the number of dimensions. If we want to specify a line, one way is to use a parametrization of a line. The parametrization of a line has the advantage that it gives you a line no matter how many dimensions you are working in.

Example 2: $x=3$

Is the graph of $x=3$ the graph of a line or a plane or a point? Just like above, the answer will depend on the dimensions we are working in. In one dimension, $x=3$ is just a point, as in a point on the number line.

In two dimensions, the graph is a line, as you can see by rotating the below graph.

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A point or a line. The graph of the equation $x=3$ could be a point, if it were an equation in one variable. But, in this case, it is an equation in two dimensions, in the variables $x$ and $y$ (where the coefficient of $y$ just happens to be zero). Rotate the figure with your mouse to see that, since it is a graph in two dimensions, the graph is really a plane. It looks best if you just rotate the figure upward, but it can get confusing because you can rotate it in any direction. You can press Home to restore the original view.

And, in three dimensions, the graph of $x=3$ is a plane.

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A vertical line or a plane. The graph of $x=3$ would be a line, if it were an equation in two dimensions, in terms of the variables $x$ and $y$, for example. However, here it is an equation in three dimensions, in terms of the variables $x$, $y$, and $z$, so that its graph is really a plane. Rotate the graph with you mouse to see this fact. Since the equation does not depend on $y$ or $z$, the plane is parallel to both the $y$ and the $z$ axis. You can press Home to restore the original view.