Two straight paths are represented by the equations $x - 3 y = 2$ and $- 2 x + 6 y = 5.$ Check whether the paths cross each other or not.

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Solution

Check whether the given pair of linear equations satisfy the conditions of unique solution.
$x - 3 y = 2$ and $- 2 x + 6 y = 5$
$\Rightarrow x - 3 y - 2 = 0 and - 2 x + 6 y - 5 = 0$
Since, the pair of equations have a unique solution, if $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{x}}$
$\Rightarrow$ Here, we have
$a_{1} = 1, b_{1} = - 3, c_{1} = - 2$
$a_{2} = - 2, b_{2} = 6, c_{2} = - 5.$

$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{1}{- 2} = - \frac{1}{2}, \frac{b_{1}}{b_{2}} = - \frac{3}{6} = - \frac{1}{2}$

$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}},$ which does not satisfy the condition of unique solution.
Hence, the two straight paths represented by the equartions $x - 3 y = 2$ and $- 2 x + 6 y = 5$ do not cross each other.

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