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Why is it true that any two points satisfying a linear equation will give you the same graph for the line represented by the equation?

I responded: If two points P, Q satisfy the same linear equation, then the two points are on the line represented by the equation. Geometrically, two points on a plane will determine a unique line. Thus the two points will give you the same graph for the line represented by the equation.

But she then asked me the following follow-on question which I need help answering:

If you pick two other points from the equation, they still lie on the same line formed by the first set of two points. Why do both pairs give the same line - why not two different lines?

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