how to find points of intersection (1 Viewer)

[fabl]

how to find points of intersection of px + qy =1 and qx + py =1

and explain why the lines intersect on line y =x ?

 

[youngsky]

Let: px + qy = qx + py (as both are equal to 1)

py - qy = px - qx
y(p-q) = x(p-q)
Cancelling common factor (p-q), you get y = x.
Hence although p and q are variable, both lines will nevertheless intersect on the line y = x.

 

[braintic]

how to find points of intersection of px + qy =1 and qx + py =1

and explain why the lines intersect on line y =x ?

Or more conventionally:

Mult 1st by p:
(p^2)x+pqy=p

Mult 2nd by q:
(q^2)x+pqy=q

Subtract:
(p^2 - q^2) x = (p-q)
(p-q)(p+q)x = (p-q)
x=1/(p+q)

Instead multiplying the 1st by q and the 2nd by p gives y = 1/(p+q)

They intersect on y=q because they are inverse relations.
That is, the roles of x and y have been swapped.