Meaning of Equation of a line

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What do we mean by "equation of a line"? How would it be defined? I know its form but what is it exactly?

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To simplify the discussion, I will treat the “line” as a “straight line”. The logic is extendable to other “conic lines”.

A line is a collection of related points with co-ordinates (x, y) in the xy-plane.

The equation of a line is a way to relate the x and y.

For example, if a line goes through A(3, 4) and B(7, 8), we want to find out the relation (i.e. the common characteristics) between the x-coordinate and the y-coordinate of some other point P(x, y) lying on the same line. In this case, P is a representative of all the points on that line. A simple guess is y – x = 1 and it shows a relation between x and y.

Some characteristics are easy to guess from the given but most of them are not. That is why we need to develop different “forms” to help us to find such relation without going through wild guessing or the process by starting from square one.