Two straight paths are represented
Question:

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

Solution:

Given linear equations are

$x-3 y-2=0$ $\ldots$ (i)

and $\quad-2 x+6 y-5=0 \quad \ldots$ (ii)

On comparing both the equations with $a x+b y+c=0$, we get

$a_{1}=1, b_{1}=-3$

and $\quad c_{1}=-2$  [from Eq. (i)]

$a_{2}=-2, b_{2}=6$

and $\quad c_{2}=-5 \quad$ [from Eq. (ii)]

Here,   $\frac{a_{1}}{a_{2}}=\frac{1}{-2}$

$\frac{b_{1}}{b_{2}}=\frac{-3}{6}=-\frac{1}{2}$ and $\frac{c_{1}}{c_{2}}=\frac{-2}{-5}=\frac{2}{5}$

i.e., $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ [pärallel lines]

Hence, two straight paths represented by the given equations never cross each other, because they are parallel to each other.