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Are those things mean the same thing or they are actually different?

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One is an algebraic formula, the other is the geometric representation of that formula.

They are the same as much as a straight line and the equation describing said line are the same.

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    $\begingroup$ One could argue that the budget constraint is an inequality, while the line is (a geometric representation of) the locus of points along which that inequality binds. $\endgroup$ – Ubiquitous Mar 12 '18 at 22:34

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