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Today, my teacher had discussed the different solutions a system of equations could possibly have (I.e. infinite solutions, single solutions, and no solutions). My teacher had also discussed how a system of equations with infinite solutions is parallel, similar to how a system of equations with no solutions is necessarily parallel. I had then asked the question, "How could a system of equations with an infinite number of solutions be possibly parallel if they always meet, for parallel lines never meet." My teacher had responded with, "Parallel lines do not necessarily not meet." I was then confused and came here to seek an answer.

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    Assuming you're not talking about projective geometry or some other weird space, parallel lines "meet" (I assume you mean intersect) iff they are coincident (meaning they're just the same line). For example $\begin{cases}2x+y=5 \\ 4x+2y=10\end{cases}$ has two equations, but both represent the exact same line. So they're parallel, intuitively meaning that they "go in the same direction", but they intersect at all of their points because they're really the same equation; the second was just multiplied by $2$ on both sides. – 2017-02-27

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A set of solutions have in common that in certain directions their vector part is 0. For a line what is common for all vectors along it is that they are not allowed to deviate from the line.

That is, any additive component of the vector that is perpendicular to the line must always be 0.

The corresponding for a 2D plane is that part of any vector in the plane can not have a non-0 additive component which points out of the plane.

So what is common amongst solutions is that they have some common part which must equal some constant.

It is possible to depict for up to 3D but for higher dimensions parameters are probably a better way to explain this. A line has 1 parameter. A plane has 2 parameters. What makes a solution unique is this combination of parameters and what makes the set of solution unique is the directions of the vectors where the parameter must be fixed ( is not allowed to vary ).