Question

Determine whether the lines are parallel, intersect, or coincide. x-5y=0, y+1=1/5(x+5)

Answer 1

Answer-

The lines are two coinciding lines or the same lines.

Solution-

The given line equations are

the first one,

$\Rightarrow x-5y=0$  

the second one,

$\Rightarrow y+1=\dfrac{1}{5}(x+5)\\\\\Rightarrow 5y+5=x+5\\\\\Rightarrow 5y=x\\\\\Rightarrow x-5y=0$

As we know two line equations $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$ will be,

1. Parallel if, $\dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}$
2. Coincide if, $\dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}=\dfrac{C_1}{C_2}$
3. Intersect if, $\dfrac{A_1}{A_2}\neq \dfrac{B_1}{B_2}$

As here,

$\Rightarrow \dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}=\dfrac{C_1}{C_2}$

$\Rightarrow \dfrac{1}{1}=\dfrac{-5}{-5}$

(C₁ and C₂ aren't considered as they are 0)

Therefore, the lines are two coinciding lines or the same lines.