Question

If $y$ varies as the sum of two quantities, one of which varies directly as $x$ and the other varies inversely as $x$; and if $y=6$ and $x=4$, and $y=3\frac{1}{3}$ when $x=3$; find the equation between $x$ and $y$.

Answer 1

According to the question let,$y=ax+\frac{b}{x}$

Now for $y=6$ we have $x=4,-1$ we get two equations as,

$4a+\frac{b}{4}=6$

and

$-a-b=6$

multiplying second equation by $4$ and adding to one we have,

$-4b+\frac{b}{4}=6+24$

$\frac{-15b}{4}=30\to b=-8$

and

$-a+8=6\to a=2$

we have $y=2x-\frac{8}{x}$