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  "body": "\n## Definition\n\nLet $F$ be a field and let $m, n \\ge 1$ be integers.\nConsider a *system of linear equations* over $F$, with $m$ equations in the unknowns $x_1, \\dots, x_n$. Such a system is a list of [[linear equations|0Q3ZT8J05R]]\n$$\\begin{aligned} a_{11} x_1 + a_{12} x_2 + \\cdots + a_{1n} x_n \u0026= b_1, \\\\ \u0026\\ \\ \\vdots \\\\ a_{m1} x_1 + a_{m2} x_2 + \\cdots + a_{mn} x_n \u0026= b_m, \\end{aligned}$$\nwhere all $a_{ij}$ and $b_i$ are elements of $F$.\nA *solution* of the system is a tuple $(s_1, \\dots, s_n) \\in F^n$ that is a solution of each of the $m$ equations.\nThe *solution set* of the system is the set of all its solutions.\n\n## Remarks\n\nWith the [[coefficient matrix|0HQFZAP3ND]] $A = (a_{ij})$, the vector $b = (b_1, \\dots, b_m)^T$ and $x = (x_1, \\dots, x_n)^T$, the system is the single equation $Ax = b$.\nOver a field, a system has no solution, exactly one solution, or more than one solution.\nWhen $F$ is infinite, \"more than one\" means infinitely many.\n\n## Explanations\n\n### L1 human\n\nA system of linear equations asks for values of the unknowns that make several linear equations true together.\nFor example, $x + y = 3$ and $x - y = 1$ together have the single solution $x = 2$, $y = 1$.\nIn the plane, each equation is a line, and a solution is a point on all the lines.\n",
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