Parametrization of linear equations
by Hidenori
Proposition
Parametrize all solutions to the linear equations
\[\begin{align*} x + 2y - 2z + w &= -1, \\ x + y + z - w &= 2. \end{align*}\]Solution
From
\[\begin{align*} \begin{bmatrix} 1 & 2 & -2 & 1 \\ 1 & 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}, \end{align*}\]we get $(x, y, z, w) = (-4t + 3s, 3t - 2s, t, s)$.
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