Question

solve xy= x+y+3 for integer values of x and y​

Answer 1

$\huge \boxed{ \sf \red{x = - 3}} \\ \huge \boxed{ \sf \blue{y = 0}}$

see the attachment for explanation.

hope it helps.

Answer 2

Topic

Diophantine equations.

-Simon's favorite factorization.

• It refers to the method of factorization by isolating unknowns and constant terms.

Solution

To solve this Diophantine equation, let's isolate unknowns by unknowns, constants by constants.

$\implies xy-x-y=3$

Adding $1$ on both sides,

$\implies (x-1)(y-1)=4$

This is called Simon's favorite factorization.

Now as the left-hand side is factorized, we can think of factorizing the right-hand side as well.

$\implies (x-1)(y-1)=4\cdot 1$

$\implies \boxed{x=5\ \text{and}\ y=2}$

$\implies (x-1)(y-1)=2\cdot 2$

$\implies \boxed{x=3\ \text{and}\ y=3}$

$\implies (x-1)(y-1)=1\cdot 4$

$\implies \boxed{x=2\ \text{and}\ y=5}$

Of course, the factors can be negative integers.

$\implies (x-1)(y-1)=(-4)\cdot (-1)$

$\implies \boxed{x=-3\ \text{and}\ y=0}$

$\implies (x-1)(y-1)=(-2)\cdot (-2)$

$\implies \boxed{x=-1\ \text{and}\ y=-1}$

$\implies (x-1)(y-1)=(-1)\cdot (-4)$

$\implies \boxed{x=0\ \text{and}\ y=-3}$

Advanced problems

Question: Solve $3x+5y=90$ over natural numbers.

Answer: $(x,y)=(5,15), (10,12), (15,9), (20,6), (25,3)$

Answer key:

Solution: From $3x=90-5y=5(18-y)$, and since 3 and 5 are co-prime, $x$ is a multiple of 5.

Let $x=5k$. Then $15k+5y=90$.

Divide both sides by 5 to obtain $3k+y=18$. Then, similarily $y=18-3k=3(6-k)$, so $y$ is a multiple of 3.

Let $y=3l$. Then $15k+15l=90$. Divide both sides by 15 to obtain $k+l=6$.

Then, solution pairs are $(k,l)=(1,5),(2,4),(3,3),(4,2),(5,1)$. And we know that $x=5k$ and $y=3l$. So the solution pairs to this equation are $\boxed{(x,y)=(5,15), (10,12), (15,9), (20,6), (25,3)}$.