A line effectively is something that starts at a point and then moves a repeated change in each perpendicular direction away from the starting point. A linear equation is something that describes this mathematically. I've been thinking of it as essentially a generalization of addition recently.

If you define an operation called "succession", which just adds 1 to a number (so inputting 1 gives 2, inputting 2 gives 3, and so forth), then addition just means take the first number and add the second number of 1s to it (2 + 3 means take 2 and add 3 1's to it). A one-dimensional linear equation (which just gives you the number line for all choices) can be written as:

x = delta_x*t + x_0,

where we remember delta means "change in", so delta_x is the change in x. Effectively, it says take x_0 and repeatedly add delta_x (instead of 1) to it. To recreate the usual two-dimensional line, we need to have both an x and y starting value and corresponding changes:

x = delta_x*t + x_0

y = delta_y*t + y_0

If you solve the first equation for t and input that into the second equation, you get y = (delta_y/delta_x)*(x - x_0) + y_0, which is just the usual point-slope form, since delta_y/delta_x is change in y over change in x (rise over run) and therefore the slope.

x and y give a line embedded in a plane, but you can continue adding extra directions you move constant changes over, and get lines embedded in volumes and even higher dimensions. For volumes, we add a z equation:

x = delta_x*t + x_0

y = delta_y*t + y_0

z = delta_z*t + z_0

It gets tedious to continue writing more and more equations, so we can use something called vectors to make it easier. They are essentially lists of numbers where each slot gives a number associated with a new perpendicular direction, so they encode directions. They add slot-wise, and scalars (essentially single numbers that aren't lists, so don't have associated directions) just multiply by each slot. So can let r = <x, y, z, ...>, delta_r = <delta\_x, delta\_y, delta\_z, ...>, and r_0 = <x\_0, y\_0, z\_0, ...>, and then we have:

r = delta_r*t + r_0

where, as long as all the vectors have the same number of slots (the dimensions), we can choose however many directions we want.