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Solve a system of equations using Cramer’s rule

by RoRi

Use Cramer’s rule to solve the system of equations:

    \begin{align*} x + y + 2z &= 4, \\ 3x - y - z &= 2, \\ 2x + 5y + 3z &= 3. \end{align*}

From Cramer’s rule we have,

    \begin{align*} x &= \frac{ \begin{vmatrix*}[r] 4 & 1 & 2 \\ 2 & -1 & -1 \\ 3 & 5 & 3 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 2 \\ 3 & -1 & -1 \\ 2 & 5 & 3 \end{vmatrix*} } = \frac{25}{25} = 1, \\[10pt] y &= \frac{ \begin{vmatrix*}[r] 1 & 4 & 2 \\ 3 & 2 & -1 \\ 2 & 3 & 3 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 2 \\ 3 & -1 & -1 \\ 2 & 5 & 3 \end{vmatrix*} } = \frac{-25}{25} = -1, \\[10pt] z &= \frac{ \begin{vmatrix*}[r] 1 & 1 & 4 \\ 3 & -1 & 2 \\ 2 & 5 & 3 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 2 \\ 3 & -1 & -1 \\ 2 & 5 & 3 \end{vmatrix*} } = \frac{50}{25} = 2. \end{align*}

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