Find the equation of line of intersection of planes: $x - y + 2 z = 5$ & $3 x + y + z = 6$

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Solution

Given equation,
$\begin{matrix} x - y + 2 z = 5 3 x + y + z = 6 x - y = 5 3 x + y = 6 - ---------- - x = \frac{11}{4}, y = \frac{- 9}{4}, z = 0 \end{matrix}$
Line of intersection is parallel to $\to r_{1} \times \to r_{2}$ .
$\begin{matrix} ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 1 & - 1 & 2 3 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ˆ i (- 3) - ˆ j (1 - 6) + ˆ k (4) = - 3 ˆ i + 5 ˆ j + 4 ˆ k \end{matrix}$
Equation of line of intesection of plane.
$\frac{x - \frac{11}{4}}{- 3} = \frac{y + \frac{9}{4}}{5} = \frac{z}{4}$

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