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@jemoka / Jemoka Knowledge Base / wiki/concepts/solving_systems.md
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--- title: "solving systems" type: concept related: [Matricies] source: https://www.jemoka.com/posts/kbhsolving_systems/ confidence: high status: active --- So let’s say given a system: \begin{equation} \begin{cases} x + 2y + z = 0 \\ 2x + 0y - z = 1 \\ x - y + 0z = 2 \end{cases} \end{equation} We can represent this using a matricies. \begin{equation} \begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} \end{equation} We will use Gaussian elimination. We will begin by multiplying the top row by \(-2\). \begin{equation} \begin{pmatrix} -2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} =\begin{pmatrix} -2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\2 \end{pmatrix} \end{equation} And then we add row one to row two; we will not write out the transformation matrix: \begin{equation} \begin{pmatrix} -2 &-4 &-2 \\ 2 &-0 &-1 \\ 1 &-1 &0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\2 \end{pmatrix} \end{equation}