Question

Solve $\frac{1}{x+y}-\frac{1}{2x}=\frac{1}{30},\frac{5}{x+y}+\frac{1}{x}=\frac{4}{3}$. Hence find the value of $2x^2-y^2$

Answer 1

Given equations are,

$\frac{1}{x+y}-\frac{1}{2x}=\frac{1}{30}$ ...(i)

$\frac{5}{x+y}+\frac{1}{3}=\frac{4}{3}$ ...(ii)

Let$(x+y)=a$

Then we have,

$\frac{1}{a}-\frac{1}{2x}=\frac{1}{30}$ ...(iii)

$\frac{5}{a}+\frac{1}{x}=\frac{4}{3}$ ...(iv)

Multiplying eq(iii) by 5

$\frac{5}{a}-\frac{5}{2x}=\frac{1}{6}$ ...(v)

Subtracting eq (iv) from eq (v)

$-\frac{5}{2x}-\frac{1}{x}=\frac{1}{6}-\frac{4}{3}$

$\Rightarrow \frac{(-5-2)}{2x}=\frac{(1-8)}{6}$

$\Rightarrow -\frac{7}{2x}=-\frac{7}{6}$

$\Rightarrow 2x=6$

$\Rightarrow x=3$

Substituting the value of $x$ in eq (iii)

$\frac{1}{a}-\frac{1}{2\times 3}=\frac{1}{30}$

$\Rightarrow \frac{1}{a}-\frac{1}{6}=\frac{1}{30}$

$\Rightarrow \frac{1}{a}=\frac{1}{30}+\frac{1}{6}$

$\Rightarrow \frac{1}{a}=\frac{1+5}{30}$

$\Rightarrow \frac{1}{a}=\frac{6}{30}$

$\Rightarrow a=\frac{30}{6}$

$\Rightarrow a=5$

Substituting the values of $a$ and $x$ in $x+y=a$, we get

$3+y=5$

$\Rightarrow y=5-3$

$\therefore y=2$

Hence $x=3,y=2$

$\therefore 2x^2-y^2=2\times 3^2-2^2$

$=2\times 9-4$

$=18-4$

$=14$

Hence the value of $2x^2-y^2$ is $14$.