e. $$ \frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x} $$

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UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x}$$
- step1: Rewrite the fractions:
  $$\frac{x^{2}}{x^{2}-y^{2}}-\frac{x}{-y+x}$$
- step2: Factor the expression:
  $$\frac{x^{2}}{\left(x+y\right)\left(x-y\right)}-\frac{x}{-y+x}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{x^{2}}{\left(x+y\right)\left(x-y\right)}-\frac{x\left(x+y\right)}{\left(-y+x\right)\left(x+y\right)}$$
- step4: Rewrite the expression:
  $$\frac{x^{2}}{\left(x+y\right)\left(x-y\right)}-\frac{x\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$$
- step5: Transform the expression:
  $$\frac{x^{2}-x\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$$
- step6: Multiply the terms:
  $$\frac{x^{2}-\left(x^{2}+xy\right)}{\left(x+y\right)\left(x-y\right)}$$
- step7: Subtract the terms:
  $$\frac{-xy}{\left(x+y\right)\left(x-y\right)}$$
- step8: Rewrite the fraction:
  $$-\frac{xy}{\left(x+y\right)\left(x-y\right)}$$
- step9: Simplify the product:
  $$-\frac{xy}{x^{2}-y^{2}}$$
The simplified form of the expression $$ \frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x} $$ is $$ -\frac{xy}{x^{2}-y^{2}} $$.
The simplified form of the expression is $$ -\frac{xy}{x^{2}-y^{2}} $$.

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