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.

Answer 1

To find whether the two lines cross each other or not we will need to find out the relationship between the coefficients of linear equations.

$x-3y=2$

$-2x+6y=5$

On comparing these two lines with the standard form of line i.e. $ax+by+c=0$, then

We have $a_1=1,b_1=-3and{c}_1=-2$

$a_2=-2,b_2=6and{c}_2=-5$

Now,

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

So, we can say that $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$, and this is the condition for the parallel lines.

And we know that the parallel lines do not cross each other.

Hence, $x-3y=2$ and$-2x+6y=5$ does not cross each other.