May one explain how the sides of the parallelogram were derived?

Question

 

https://openstax.org/books/calculus-volume-3/pages/5-7-change-of-variables-in-multiple-integrals

Answer 1

Left side of parallelogram passes through (1,2) (3,4) so passes through (x1, y1), (x2,y2) where

x1 = 1, y1 = 2, x2 = 3, y1 = 4

Equation of this line is (y-y1)/ (x-x1) = (y2-y1) / (x2-x1)

So substituting

(y-2) / (x-1) = (4-2) / (3-1) = 2/2 = 1

So (y-2) / (x-1) = 1

So y-2 = x-1

So x-y+1 = 0 as stated above

You can use the same method for the other 3 sides.

MIKE