Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\frac{x+y}{(x+y)^{2}-z^{2}}+\frac{y-z}{x^{2}-(y-z)^{2}}-\frac{z+x}{(z+x)^{2}-y^{2}}$$

Answer

((x+y)*(-y+x+z)+(y-z)*(x+y+z)-(z+x)*(x+y-z))/((x+y+z)*(x+y-z)*(-y+x+z))

Solution


Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{x+y}{(x+y+z)(x+y-z)}+\frac{y-z}{{x}^{2}-{(y-z)}^{2}}-\frac{z+x}{{(z+x)}^{2}-{y}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{x+y}{(x+y+z)(x+y-z)}+\frac{y-z}{(x+y-z)(x-(y-z))}-\frac{z+x}{{(z+x)}^{2}-{y}^{2}}\]
Remove parentheses.
\[\frac{x+y}{(x+y+z)(x+y-z)}+\frac{y-z}{(x+y-z)(x-y+z)}-\frac{z+x}{{(z+x)}^{2}-{y}^{2}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{x+y}{(x+y+z)(x+y-z)}+\frac{y-z}{(x+y-z)(x-y+z)}-\frac{z+x}{(z+x+y)(z+x-y)}\]
Simplify.
\[\frac{x+y}{(x+y+z)(x+y-z)}+\frac{y-z}{(x+y-z)(-y+x+z)}-\frac{z+x}{(x+y+z)(-y+x+z)}\]
Rewrite the expression with a common denominator.
\[\frac{(x+y)(-y+x+z)+(y-z)(x+y+z)-(z+x)(x+y-z)}{(x+y+z)(x+y-z)(-y+x+z)}\]
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