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Q. 76

Expert-verifiedFound in: Page 774

Book edition
6th

Author(s)
Sullivan

Pages
1200 pages

ISBN
9780321795465

The sum of two numbers is the same as their product, and the difference of their reciprocals is $3$. Find the numbers.

The two numbers are $\frac{1}{2}and-1$.

Assume the two numbers to be *x *and *y*.

From the question,

$x+y=xy......\hspace{0.17em}\left(i\right)\phantom{\rule{0ex}{0ex}}\frac{1}{x}-\frac{1}{y}=3......\left(ii\right)$

Multiply equation (ii) with *xy,*

role="math" localid="1646841503959" $\frac{xy}{x}-\frac{xy}{y}=3xy\phantom{\rule{0ex}{0ex}}y-x=3xy......\left(iii\right)$

Add equations (i) and (iii),

$\left(x+y\right)+\left(y-x\right)=xy+3xy\phantom{\rule{0ex}{0ex}}2y=4xy\phantom{\rule{0ex}{0ex}}x=\frac{1}{2}$

Substitute the value of *y *equation (i),

$\frac{1}{2}+y=\frac{1}{2}y\phantom{\rule{0ex}{0ex}}y-\frac{y}{2}=-\frac{1}{2}\phantom{\rule{0ex}{0ex}}y=-1$

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