Question

Felipe thinks $\frac{1}{x}+\frac{1}{y}$is $\frac{2}{x+y}$.

ⓐ Choose numerical values for $x$and $y$and evaluate $\frac{1}{x}+\frac{1}{y}$.

ⓑ Evaluate $\frac{2}{x+y}$for the same values of $x$and $y$you used in part (a).

ⓒ Explain why Felipe is wrong.

ⓓ Find the correct expression for $\frac{1}{x}+\frac{1}{y}$.

Short answer

Part a. For $x=1$and $y=1$, the value of the expression $\frac{1}{x}+\frac{1}{y}$is $2$.

Part b. For the same values, the value of the expression $\frac{2}{x+y}$is $1$.

Part c. As the values of part (a) and part (b) are different so Felipe is wrong.

Part d. The correct expression for$\frac{1}{x}+\frac{1}{y}$is$\frac{x+y}{xy}$.

Step-by-step solution

Part (a) Step 1. Given Information 

It is given that Felipe thinks$\frac{1}{x}+\frac{1}{y}$is $\frac{2}{x+y}$.

Part (a) Step 2. Find the value of 1x+1y

Let us take the value of $x$and $y$both to be $1$.

For these values, the value of the expression $\frac{1}{x}+\frac{1}{y}$is given as

$\begin{array}{rcl}\frac{1}{x}+\frac{1}{y}& =& \frac{1}{1}+\frac{1}{1}\\ & =& 1+1\\ & =& 2\end{array}$

Part (b) Step 1. Find the value of 2x+y

In the expression $\frac{2}{x+y}$, substitute $1$for $x$and $1$for $y$

$\begin{array}{rcl}\frac{2}{x+y}& =& \frac{2}{1+1}\\ & =& \frac{2}{2}\\ & =& 1\end{array}$

Part (c) Step 1. Felipe is wrong

For $x=1$and $y=1$, we get different values for the expressions $\frac{1}{x}+\frac{1}{y}$and $\frac{2}{x+y}$. So Felipe is wrong that$\frac{1}{x}+\frac{1}{y}$is$\frac{2}{x+y}$.

Part (d) Step 1. Add the fractions

For the fractions $\frac{1}{x}+\frac{1}{y}$, the LCD is $xy$. And so the fractions can be added as

$\begin{array}{rcl}\frac{1}{x}+\frac{1}{y}& =& \frac{1·y}{x·y}+\frac{1·x}{y·x}\\ & =& \frac{y}{xy}+\frac{x}{xy}\\ & =& \frac{x+y}{xy}\end{array}$