Solve $5 x - 6 y + 4 z = 157 x + 4 y - 3 z = 192 x + y + 6 z = 46$ .

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Solution

Here $D = ∣ ∣ ∣ ∣ \begin{matrix} 5 & - 6 & 4 7 & 4 & - 3 2 & 1 & 6 \end{matrix} ∣ ∣ ∣ ∣$
Apply $C_{1} - 2 C_{1}, C_{3} - 6 C_{2}$
$∴ D = ∣ ∣ ∣ ∣ \begin{matrix} 17 & - 6 & 40 - 1 & 4 & - 27 0 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ = - ∣ ∣ ∣ \begin{matrix} 17 & 40 - 1 & - 27 \end{matrix} ∣ ∣ ∣ = - [- 459 + 40] = 419 \neq 0$
Since $D \neq 0$ , therefore the equations are consistent.
$D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 15 & - 6 & 4 19 & 4 & - 3 46 & 1 & 6 \end{matrix} ∣ ∣ ∣ ∣ = 15 (27) - 19 (- 40) + 46 (2) = 1257$
Similarly, $D_{2} = 1676, D_{3} = 2514$
$∴$ $\frac{x}{1257} = \frac{y}{1676} = \frac{z}{2514} = \frac{1}{419} =∴ x = 3, y = 4, z = 6$

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