Solving Algebraic Fractions

I have tried to work out the following questions, but I am not sure if my answer is correct The question says to simplify..

What I basically did for the first one is combine the denominators and numerators, and multiplied them together to find the lowest common factor.

For the second question, I inverted the second part of the equation to make it a multiplication sign, then I just factored it out and cancelled anything I could.

Thanks, regards.

• If you just want to check your answer, just use Wolfram Alpha... – fgp Mar 31 '14 at 11:40
• Asking Wolfram Alpha about the first expression yields this – fgp Mar 31 '14 at 11:42
• WA won't help many during an exam...and many times WA yields wrong/incomplete solutions. – DonAntonio Mar 31 '14 at 11:54

First recommended step: simplify as much as possible each numerator and denominator in each summand

$$\frac{x^2y^2}{x^2y-xy^2}+\frac{x^2(x+y)}{x^2-y^2}=\frac{\color{red}x^2\color{red}y^2}{\color{red}{xy}(x-y)}+\frac{x^2\color{red}{(x+y)}}{(x-y)\color{red}{(x+y)}}=$$

$$=\frac{xy}{x-y}+\frac{x^2}{x-y}\;\;\;\;(**)$$

Now, since the above happens to be the sum of two fractions with a common denominator, just...sum them:

$$(**)=\frac{xy+x^2}{x-y}=\frac{x(x+y)}{x-y}$$

For the first one directly do the fractions' (check carefully the minimal common denominator):

$$\frac y{x(x-y)}+\frac1x=\frac{y+x-y}{x(x-y)}=\frac x{x(x-y)}=\frac1{x-y}$$

In case the second equation has a division symbol instead of a plus one:

$$\frac{x^2y^2}{x^2y-xy^2}\div\frac{x^2(x+y)}{x^2-y^2}=\frac{\color{red}x^2\color{red}y^2}{\color{red}{xy}(x-y)}\div\frac{x^2\color{red}{(x+y)}}{(x-y)\color{red}{(x+y)}}=$$

$$=\frac{xy}{x-y}\div\frac{x^2}{x-y}\;\;(**)$$

And we now use the basic rule $\;\frac ab\div\frac cd=\frac ab\cdot \frac dc\;$ :

$$(**)=\frac{\color{green}xy}{\color{red}{x-y}}\cdot\frac{\color{red}{x-y}}{\color{green}x^2}=\frac yx$$

• I think OP's second expression has a division sign not addition. – user103816 Mar 31 '14 at 11:58
• Hmmm...perhaps you're right, @Anupam . Let us wait until the OP addresses this matter. Thanks. – DonAntonio Mar 31 '14 at 12:01

For first expression you can try taking $\dfrac{1}{x}$ common in the denominator. $$\dfrac{y}{x^2-xy}+\dfrac{1}{x}=\dfrac{1}{x}(\dfrac{y}{x-y})+\dfrac{1}{x}=\dfrac{1}{x}(\dfrac{y}{x-y}+1)=\dfrac{1}{x}(\dfrac{y+x-y}{x-y})=\dfrac{1}{x-y}$$

For the second expression first take out common things apart, that is
$$\dfrac{{(xy)}^2}{x^2y-xy^2}=\dfrac{{(xy)}^2}{xy(x-y)}$$ And
$$\dfrac{x^2(x+y)}{x^2-y^2}=\dfrac{x^2(x+y)}{(x-y)(x+y)}$$
And then proceed. The result of the first expression should come out $\dfrac{1}{x-y}$ and the result of the second should come out $\dfrac{y}{x}$.