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Using Cramer’s rule, solve the given system of equations

by RoRi

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

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

From Cramer’s rule we have,

    \begin{align*} x &= \frac{ \begin{vmatrix*}[r] 5 & 1 & 0 \\ 2 & 0 & 1 \\ 5 & 1 & 1 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{vmatrix*} } = \frac{-2}{-2} = 1 \\[10pt] y &= \frac{ \begin{vmatrix*}[r] 1 & 5 & 0 \\ 1 & 2 & 1 \\ 0 & 5 & 1 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{vmatrix*} } = \frac{-8}{-2} = 4 \\[10pt] z &= \frac{ \begin{vmatrix*}[r] 1 & 1 & 5 \\ 1 & 0 & 2 \\ 0 & 1 & 5 \end{vmatrix*} }{ \begin{vmatrix*}[r] 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{vmatrix*} } = \frac{-2}{-2} = 1. \end{align*}

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