Solve the following simultaneous equations using Cramer’s rule.
3x - 4y = 10 ; 4x + 3y = 5

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Solution

$D = ∣ ∣ ∣ \begin{matrix} 3 & - 4 4 & 3 \end{matrix} ∣ ∣ ∣ = 3 \times 3 - (- 4) \times 4 = 9 + 16 = 25$
$D_{x} = ∣ ∣ ∣ \begin{matrix} 10 & - 4 5 & 3 \end{matrix} ∣ ∣ ∣ = 10 \times 3 - (- 4) \times 5 = 30 + 20 = 50$
$D_{y} = ∣ ∣ ∣ \begin{matrix} 3 & 10 4 & 5 \end{matrix} ∣ ∣ ∣ = 3 \times 5 - 10 \times 4 = 15 - 40 = - 25$
$x = \frac{D_{x}}{D} = \frac{50}{25} = 2$
$y = \frac{D_{y}}{D} = \frac{- 25}{25} = - 1$
$(x, y) = (2, - 1)$

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