How can I represent an N dimensional line?

How can I represent a straight line (between two points) in a N-dimensional space?

If $x,y$ are in $n$-dimensional space the line between them can be represented by $t(x-y)+y$ where $t$ runs over the real numbers. For a sanity check, observe that we get $y$ when $t=0$ and $x$ when $t=1$. You can't write a line as the solutions to a single equation in $n$ dimensions, unfortunately-you need $n-1$ equations, which is why you do get $y=mx+b$ etc in $2$ dimensions.

You can take n-1 linearly independent hyperplanes which go through these two points and solve the system formed by those equations. The result will be the equation of the line.

I suppose that the form of the equation of the line could be: $$\frac{x_1-x^0_1}{a_1}=\frac{x_2-x^0_2}{a_2}=...\frac{x_n-x^0_1}{a_n}$$ where $(x^0_1,x^0_2,...x^0_n)$ is a given point of the line and $(a_1,a_2,...,a_n)$ a vector colinear to the line.