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  "body": "\n## Definition\n\nLet $F$ be a field.\nA [[system of linear equations|0GT4WWGWH2]] over $F$ is *consistent* when it has at least one solution in $F^n$.\nIt is *inconsistent* when it has no solution.\n\n## Remarks\n\nThe system $x + y = 1$, $x + y = 2$ over $\\R$ is inconsistent.\nEvery homogeneous system is consistent, because the zero tuple is a solution.\nThe [[Rouché–Capelli theorem|0HV1SKWPMS]] gives a test with ranks: $Ax = b$ is consistent if and only if $\\rank A = \\rank [A \\mid b]$.\n\n## Explanations\n\n### L1 human\n\n\"Consistent\" means that the equations can all be true at the same time.\nTwo parallel lines in the plane, like $x + y = 1$ and $x + y = 2$, have no common point, so their system is inconsistent.\nElimination shows inconsistency when it produces an equation $0 = c$ with $c \\ne 0$.\n",
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