Question

Solve the following system of equations.
$\frac{1}{x}-\frac{1}{y}=\frac{2}{15},{\log }_{3}x+{\log }_{3}y=1+{\log }_{3}5$

Answer 1

$\frac{1}{x}-\frac{1}{y}=\frac{2}{15}{\log }_{3}x+{\log }_{3}y=1+{\log }_{3}5$

${\log }_3\left(xy\right)={\log }_{3}3+{\log }_{3}5$

$\frac{y-x}{xy}=\frac{2}{15}\to 1{\log }_3\left(xy\right)={\log }_3\left(15\right)$

$xy=15\to 2$

From $1$ and $2$

$(\frac{15}{x}-x)=\frac{2}{15}\left(15\right)$

$15-x^2=2x$

$x^2+2x-15=0$

$(x+5)(x-3)=0$

$x=3,-5$

From $2$ $xy=15$

$3y=15-5y=15$

$y=5y=-3$

$(x,y):(3,5),(-5,-3)$