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Solve a given system of equations using Cramer’s rule

by RoRi

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

    \begin{align*} x + 2y + 3z &= 5,\\ 2x - y + 4z &= 11,\\ -y + z &= 3. \end{align*}

From Cramer’s rule we have

    \begin{align*} x &= \frac{ \begin{vmatrix*}[r] 5 & 2 & 3 \\ 11 & -1 & 4 \\ 3 & -1 & 1 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 2 & 3 \\ 2 & -1 & 4 \\ 0 & -1 & 1 \end{vmatrix*} } = \frac{-7}{-7} = 1\\[10pt] y &= \frac{ \begin{vmatrix*}[r] 1 & 5 & 3 \\ 2 & 11 & 4 \\ 0 & 3 & 1 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 2 & 3 \\ 2 & -1 & 4 \\ 0 & -1 & 1 \end{vmatrix*} } = \frac{7}{-7} = -1 \\[10pt] z & = \frac{ \begin{vmatrix*}[r] 1 & 2 & 5 \\ 2 & -1 & 11 \\ 0 & -1 & 3 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 2 & 3 \\ 2 & -1 & 4 \\ 0 & -1 & 1 \end{vmatrix*} } = \frac{-14}{-7} = 2. \end{align*}

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