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# 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.

Given:

The given system of equations is:

\( x-3 y=2 \) and \( -2 x+6 y=5 \)

To do:

We have to find whether the paths represented by the given equations cross each other or not.

Solution:

The given system of equations can be written as:

$x - 3y -2=0$

$-2x +6y -5=0$

The standard form of system of equations of two variables is $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y-c_{2}=0$.

Comparing the given system of equations with the standard form of equations, we have,

$a_1=1, b_1=-3, c_1=-2$ and $a_2=-2, b_2=6, c_2=-5$

Therefore,

$\frac{a_1}{a_2}=\frac{1}{-2}=\frac{-1}{2}$

$\frac{b_1}{b_2}=\frac{-3}{6}=\frac{-1}{2}$

$\frac{c_1}{c_2}=\frac{-2}{-5}=\frac{2}{5}$

Here,

$\frac{a_{1}}{a_{2}} =\frac{b_{1}}{b_{2}} ≠ \frac{c_{1}}{c_{2}}$

This implies, the given lines are parallel to each other.

**Hence, the two straight paths represented by the given equations never cross each other.**

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