Chapter 6.1, Problem 9WE

To determine

To calculate: The simplified form of the expression $\frac{2xy}{x^{2}y-y^{2}x}$ and state restrictions on variable.

Answer to Problem 9WE

The simplified value of the expression $\frac{2xy}{x^{2}y-y^{2}x}$ is $\frac{2}{x-y}$ and restriction is $x\ne y,x\ne 0,y\ne 0$.

Explanation of Solution

Given information:

The expression $\frac{2xy}{x^{2}y-y^{2}x}$.

Formula used:

The simplest form of an algebraic fraction is when numerator and denominator have no common factor other than 1 and $-1$ . Simplification is done by factorizing numerator and denominator.

Calculation:

Consider the expression $\frac{2xy}{x^{2}y-y^{2}x}$.

Recall that the simplest form of an algebraic fraction is when numerator and denominator have no common factor other than 1 and $-1$ . Simplification is done by factorizing numerator and denominator.

So, take out $xy$ common from denominatorand then strike out common terms and simplify it as,

  $\begin{array}{c}\frac{2xy}{x^{2}y-y^{2}x}=\frac{2xy}{xy\left(x-y\right)}\\ =\frac{2}{x-y}\end{array}$

To restrict the variable, exclude the value that makes the denominator zero.

So, denominator will be zero when,

  $\begin{array}{c}{x}^{2}y-y^{2}x=0\\ xy\left(x-y\right)=0\end{array}$

There are two possibilities either $\begin{array}{c}xy=0\\ x=0\text{ or }y=0\end{array}$ or $\begin{array}{c}x-y=0\\ x=y\end{array}$.

For denominator not to be zero, $x\ne y,x\ne 0,y\ne 0$.

Thus, the simplified value of the expression $\frac{2xy}{x^{2}y-y^{2}x}$ is $\frac{2}{x-y}$ and restriction is $x\ne y,x\ne 0,y\ne 0$.