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  "body": "\n## Summary\n\nFourier–Motzkin elimination removes one variable from a system of linear inequalities, as Gaussian elimination does for equations.\nThe new system has a solution exactly when the old one has, and its solutions are the projections of the old solutions.\nRepeating it decides whether a system of linear inequalities is solvable.\nIt gives short proofs of [[Farkas' lemma|FYZC634H2D]] and of the fact that projections of polyhedra are polyhedra.\nThe number of inequalities can grow very fast, so it is mainly a theoretical tool.\n\n## Explanations\n\n### L1 human\n\nTo get rid of $y$, sort the inequalities into those that bound $y$ from above, those that bound it from below, and those without $y$.\nA value of $y$ exists exactly when every lower bound is at most every upper bound.\nSo replace all inequalities with $y$ by the comparisons \"lower bound $\\le$ upper bound\".\nFor $0 \\le y \\le 1 - x$ this gives $0 \\le 1 - x$, that is $x \\le 1$.\n",
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