Question

Given that both $x$ and $y$ are positive integers. Solve the following systems of equations.
$x^2+xy=15,y^2+xy=10$

Answer 1

Given that $x^2+xy=15$.........(1) ,$y^2+xy=10$.........(2)

add both equation (1) and (2)

$x^2+xy+y^2+xy=15+10$

$x^2+y^2+2xy=25$

$(x+y{)}^2=25$

$(x+y)=5$

Now subtract equation 2 from 1 we have

$x^2+xy-y^2-xy=15-10$

$(x+y)(x-y)=5$

$5(x-y)=5$

$(x-y)=1$

Now we have

$(x+y)=5$

and$(x-y)=1$

so add both the equation we have

$2x=6$

$x=3$

$x+y=5$

$y=2$

$x=3andy=2$

hence this the the required solution