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We know $y=mx+c$ is the equation for a straight line in a 2D graph. And the parallel line that goes through $(x_1,y_1)$ is $y=mx+(-mx_1+y_1)$. But how do we display the straight line in a 3D graph from an equation? How does we identify the line which is parallel in all 3 axis that goes through point $(x_1,y_1,z_1)$?

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A line in $\mathbb R^3$ (3d real space) can't be represented by a single equation. The reason for this is that a line is one-dimensional whereas space is 3-dimensional. The goal of writing a line in three-dimensional space is to eliminate two of these dimensions. To do this we need two equations: one to eliminate each extra dimension.

A little bit less abstractly, a line in $\mathbb R^3$ is just the intersection of two planes. This should be fairly intuitive, just imagine two infinite non-parallel planes and look at where they cross. So if we want to represent a line algebraically, all we need is a system of two plane equations. The equation of a plane is

$$ax+by+cz = r$$

where $a,b,c,r$ are constants. Now we can get our representation of a line by taking the solution set a system of two of these:

$$\begin{cases}a_1x+b_1y+c_1z=r_1\\a_2x+b_2y+c_2z=r_2\end{cases}$$

The solutions to this will form a line. As a concrete example, the intersection of the simple planes $x = 0$ and $y = 0$ will form the line that is the $z$ axis.

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Using vector notation it is straightforward to represent a line, in any number of dimensions.

$2-$dimensional: $(x, y) = (5, 3) + m (1, 0)$

$3-$dimensional: $(x, y, z) = (5, 3, 1) + m (1, 0, 0)$

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