Question

If $y$ is equal to the sum of two quantities one of which varies as $x$ directly, and the other as $x^2$ inversely; and if $y=19$ when $x=2$ or $3$; find $y$ in terms of $x$.

Answer 1

According to question let,

$y=ax+\frac{b}{x^2}$

Now as for $y=19$ we have $x=2,3$, we get two equations as

$2a+\frac{b}{4}=19$

and

$3a+\frac{b}{9}=19$

multiplying first equation by $3$ and second by $2$ and subtracting two from one we get,

$\frac{3b}{4}-\frac{2b}{9}=57-38$

$\frac{27b-8b}{36}=19$

$\frac{19b}{36}=19\Rightarrow b=36$

and

$2a+\frac{36}{4}=19\Rightarrow 2a=10\Rightarrow a=5$

The equation is $y=5x+\frac{36}{x^2}$