Solve the system of equations $x + y + 2 z = 4; 2 x + 2 y + 4 z = 8; 3 x + 3 y + 6 z = 10$ by using determinant

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Solution

Given equations are $x + y + 2 z = 4; 2 x + 2 y + 4 z = 8$ and $3 x + 3 y + 6 z = 10$ .
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 2 2 & 2 & 4 3 & 3 & 6 \end{matrix} ∣ ∣ ∣ ∣ = 0 [∵ C_{1} \equiv C_{2}]$
$Δ_{x} = ∣ ∣ ∣ ∣ \begin{matrix} 4 & 1 & 2 8 & 2 & 4 10 & 3 & 6 \end{matrix} ∣ ∣ ∣ ∣ = 2 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 1 & 1 8 & 2 & 2 10 & 3 & 3 \end{matrix} ∣ ∣ ∣ ∣$
[Taking $2$ common from $C_{3}$ ]
$= 2 (0) = 0 [∵ C_{2} \equiv C_{3}]$
$Δ_{y} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 4 & 2 2 & 8 & 4 3 & 10 & 6 \end{matrix} ∣ ∣ ∣ ∣ = 2 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 4 & 1 2 & 8 & 2 3 & 10 & 3 \end{matrix} ∣ ∣ ∣ ∣$
[Taking $2$ common from $C_{3}]$
$= 2 (0) = 0 [∵ C_{1} \equiv C_{3}]$
$Δ_{z} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 4 2 & 2 & 8 3 & 3 & 10 \end{matrix} ∣ ∣ ∣ ∣ = 0$ .... $[∵ C_{1} \equiv C_{2}]$
$∴ △ = 0$ and $△_{x} = △_{y} = △_{z} = 0$
Also all $2 \times 2$ minors of $△ = 0$ , but not all the minors of $△_{x}, △_{y}, △_{z}$ are zero.
$∴$ the system is inconsistent and has no solution.

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