Question

Complete the following table to draw the graph of 2 − 6 = 3

Solution .

x -5 ___

y. 0

(x, y. ___ ___​

Answer 1

For x,y∈Z+x,y∈Z+ we have

xy+y+1x=4⟹x2−4xy+y2+y=0⟹x=2y±3y2−y−−−−−−√⟹

xy+y+1x=4⟹x2−4xy+y2+y=0⟹x=2y±3y2−y⟹

⟹∃z∈Z+(z2=3y2−y=y(3y−1))⟹

⟹∃z∈Z+(z2=3y2−y=y(3y−1))⟹

⟹∃a,b∈Z+(y=a2∧3y−1=b2)⟹

⟹∃a,b∈Z+(y=a2∧3y−1=b2)⟹

⟹∃a,b∈Z+(3a2−1=b2)⟹

⟹∃a,b∈Z+(3a2−1=b2)⟹

⟹∃b∈Z+(b2≡−1mod3).

⟹∃b∈Z+(b2≡−1mod3).

The 1st (displayed) line uses the Quadratic Formula.

In the 2nd line, z≠0z≠0 because y∈Z+⟹y(3y−1)>0.y∈Z+⟹y(3y−1)>0.

In the 3rd line, aa and bb exist because yy and 3y−13y−1 are co-prime members of Z+Z+ and their product is the square of the positive integer