Question

Show that the lines $2x+5y=1,x-3y=6$ and $x+5y+2=0$ are concurrent

Answer 1

Three lines are concurrent if they pass through the same point.

Given,

$L_1:2x+5y=1...\left(1\right)$

$L_2:x+5y+2=0...\left(2\right)$

$L_3:x-3y=6...\left(3\right)$

Solving equations $\left(1\right)$ and $\left(2\right)$, we get the intersection point of lines $L_1$ and $L_2$

Subtracting equation $\left(2\right)$ from $\left(1\right),$

$2x+5y-(x+5y+2)=1-0\Rightarrow x-2=1\Rightarrow x=3$

Now putting the value of x in equation 1 we have,

$2\times 3+5y=1\Rightarrow 5y=1-6\Rightarrow 5y=-5\Rightarrow y=-1$

So the intersection point of lines $L_1$ and $L_2$ is $(3,-1)$

Putting the values of $x$ and $y$ in equation $3$ we have,

$3-3(-1)=6\Rightarrow 3+3=6$

So, the lines $L_1,L_2$ and $L_3$ all pass through the point $(3,-1)$

Thus, these lines are concurrent.