Question

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

Short answer

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

Step-by-step solution

Step 1. Given information.

Assume the two numbers to be xand y.

From the question,

$$\begin{gathered} x+y=xy......\hspace{0.17em}\left(i\right) \\ \frac{1}{x}-\frac{1}{y}=3......\left(ii\right) \end{gathered}$$

Multiply equation (ii) with xy,

role="math" localid="1646841503959" $$\begin{gathered} \frac{xy}{x}-\frac{xy}{y}=3xy \\ y-x=3xy......\left(iii\right) \end{gathered}$$

Step 2. Find the two numbers.

Add equations (i) and (iii),

$$\begin{gathered} \left(x+y\right)+\left(y-x\right)=xy+3xy \\ 2y=4xy \\ x=\frac{1}{2} \end{gathered}$$

Substitute the value of yequation (i),

$$\begin{gathered} \frac{1}{2}+y=\frac{1}{2}y \\ y-\frac{y}{2}=-\frac{1}{2} \\ y=-1 \end{gathered}$$