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ax+by+c=0

dx+ey+f=0

We often find the point of intersection by substitution or elemination.

If we were to solve by elemination, we either add or substract the two equations (or manipulate them) to eleminate one variable, in order to express the other variable only in terms of non-variable (that is, fixed point).

What does it mean? How does a set of element that satisfies a equation be manipulated to find a subset of the set of which the subset also satisfies the other equation?

Or what does adding to equation mean 'graphically'? (I am certain that a better word exists)

I somehow can feel that adding or subtracting a non-variable to a equation, but I don't understand the significance of adding or subtracting equations