We have

4x + 6y = 3xy

and 8x + 9y = 5xy

where x≠0 and y≠0

Lets simplify these equations.

4x + 6y = 3xy

Dividing the equation by xy throughout,

⇒

Assuming p = 1/y and q = 1/x, we get

4p + 6q = 3 …(i)

Also, 8x + 9y = 5xy

Dividing the equation by xy throughout,

⇒

Assuming p = 1/y and q = 1/x, we get

⇒ 8p + 9q = 5 …(ii)

To solve these equations, we need to make one of the variables (in both the equations) have same coefficient.

Lets multiply eq.(i) by 2, so that variable p in both the equations have same coefficient.

Recalling equations (i) & (ii),

4p + 6q = 3 [×2

8p + 9q = 5

⇒ 3q = 1

⇒ q = 1/3

Substitute q = 1/3 in eq.(i)/eq.(ii), as per convenience of solving.

Thus, substituting in eq.(i), we get

4p + 6(1/3) = 3

⇒ 4p + 2 = 3

⇒ 4p = 3 – 2 = 1

⇒ p = 1/4

Thus, p = 1/4 and q = 1/3

As q = 1/x,

⇒ 1/3 = 1/x

⇒ x = 3

And p = 1/y

⇒ 1/4 = 1/y

⇒ y = 4

Hence, we have x = 3 and y = 4