Question

A motorcyclist is moving along the line $x-y=2$and another motorcyclist is moving along the line $x-y=4.$ Find out their moving direction.

Answer 1

The Given pair of linear equation is :
$$\begin{gathered} x-y=2 \\ x-y-2=0\dots \dots \dots \left(1\right) \\ x-y=4 \\ x-y-4=0\dots \dots \dots \dots \left(2\right) \end{gathered}$$
On comparing with standard form of pair of linear equations $a_{1}x+b_{1}y+c_1=0and{a}_{2}x+b_{2}y+c_2=0$
$$\begin{gathered} a_1=1,b_1=-1,c_1=-2 \\ {a}_2=1,b_2=-1,c_2=-4 \end{gathered}$$
Now,

$\frac{a_1}{a_2}=\frac{1}{1},\frac{b_1}{b_2}=\frac{-1}{-1}=\frac{1}{1},\frac{c_1}{c_2}=\frac{-2}{-4}=\frac{1}{2}$

Since,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Thus is the criteria of parallel lines.

Hence, the lines representing the pair of linear equations are parallel lines.

Thus, the motorcyclists are moving parallel to each other either in the same direction or in opposite directions.