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Let $\mathbb{K}$ be a field (in general $\mathbb{R}$ or $\mathbb{C}$). A linear equation with coefficients in $\mathbb{K}$ is an expression of the form

$$a_1x_1+\cdots+a_nx_n=b$$

where $a_i$ and $b$ are known numbers in $\mathbb{K}$ (coefficients and independent term), and $x_i$ are the unknowns, for which some values in $\mathbb{K}$ satisfy the equation (solutions) and others don't

We may impose the fulfillment of not just one, but many (say $m$) linear equations: this is a system of linear equations

$$ \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. $$

whose solutions are the values $x_i$ that satisfy the $m$ equations simultaneously