The equation ot the line passing through the point $( 1 , - 2,3 )$ and parallel to the line

$x - y + 2 z = 5$ and $3 x + y + z = 6$ is

$\begin{array}{l} \vec { r } =\vec { a } +\lambda \vec { b } \\ \Rightarrow \vec{r} = \hat { i } -2\hat { j } +3\hat { k } +\lambda \left( { \hat { i } -j+2\hat { k } } \right) \\ \Rightarrow x\hat { i } +y\hat { j } +z\hat { k } =\left( { 1+\lambda } \right) \hat { i } +\left( { 2-\lambda } \right) \hat { j } +\left( { 3+2\lambda } \right) \hat { k } \\ \Rightarrow x=1+\lambda \\ \Rightarrow y=2-\lambda \\ \Rightarrow z=3+2\lambda \\ Then,\, \, cartesion\, \, equation\, \, of\, \, the\, \, line \\ \frac { { x-1 } }{ 1 } =\frac { { y+2 } }{ { -1 } } =\frac { { z-3 } }{ 2 } \end{array}$