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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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