<s>[INST] Equation of the line passing through the intersection of two lines and is parallel to another line. The Question is :
Find the equation of the line through the intersection of the lines $3x+2y−8=0,5x−11y+1=0$ and parallel to the line $6x+13y=25$
Here is how I did it..
$L_1 = 3x + 2y -8 = 0$
$L_2 = 5x -11y +1 = 0$
$L_3= ?$
$L_4 = 6x + 13y -25= 0$
I found the point of intersection : $(-2, -1)$
Using formula: $L_1 + kL_2$ = 0
$(3x + 2y -8) + k(5x -11y +1)=0$
$(3(2) + 2(1) -88) + k(5(2) -11(1) +1) = 0$
$(6 +2 -8) + k(10 -11 +1) =0$
$8-8 + k(11-11) =0$
$0 +k(0) = 0$
What's wrong ?
 [/INST] What;'s wrong is that $(-2, -1)$ is not on the line 
$$
5x - 11y + 1 = 0
$$
because
$$
5(-2) - 11(-1) + 1 = -10 + 11 + 1 = 2. 
$$
</s>