# Simplifying Rational Expressions| Algebra| ENGG10 LE2 Problem 1

#### $\frac{\frac{\frac{ax+ay+3a}{x^2+2xy-15y^2}}{\frac{a}{x+5y}-\frac{cx-2cy-3ay}{x^2-25y^2}}}{\frac{2x^2+4xy+2y^2+5x-3+5y}{ax+3cy-3ay-cx}}$

SOLUTION:

$\frac{\frac{\frac{ax+ay+3a}{x^2+2xy-15y^2}}{\frac{a}{x+5y}-\frac{cx-2cy-3ay}{x^2-25y^2}}}{\frac{2x^2+4xy+2y^2+5x-3+5y}{ax+3cy-3ay-cx}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{a}{x+5y}-\frac{cx-2cy-3ay}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{\left(2x^2+4xy+2y^2\right)+\left(5x+5y\right)-3}{\left(ax-cx\right)+\left(3cy-3ay\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{a\left(x-5y\right)-\left(cx-2cy-3ay\right)}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{2\left(x^2+2xy+y^2\right)+5\left(x+y\right)-3}{x\left(a-c\right)-3y\left(a-c\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{ax-5ay-cx+2cy+3ay}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{2\left(x+y\right)^2+5\left(x+y\right)-3}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{ax-2ay-cx+2cy}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{\left[2\left(x+y\right)-1\right]\left[\left(x+y\right)+3\right]}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{\left(ax-cx\right)-\left(2ay-2cy\right)}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{\left(2x+2y-1\right)\left(x+y+3\right)}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{x\left(a-c\right)-2y\left(a-c\right)}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{\left(2x+2y-1\right)\left(x+y+3\right)}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}}{\frac{\left(x-2y\right)\left(a-c\right)}{\left(x+5y\right)\left(x-5y\right)}}}{\frac{\left(2x+2y-1\right)\left(x+y+3\right)}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{\frac{a\left(x+y+3\right)}{\left(x+5y\right)\left(x-3y\right)}\cdot \frac{\left(x+5y\right)\left(x-5y\right)}{\left(x-2y\right)\left(a-c\right)}}{\frac{\left(2x+2y-1\right)\left(x+y+3\right)}{\left(x-3y\right)\left(a-c\right)}}$

$=\frac{a\left(x+y+3\right)\left(x+5y\right)\left(x-5y\right)}{\left(x+5y\right)\left(x-3y\right)\left(x-2y\right)\left(a-c\right)}\cdot \frac{\left(x-3y\right)\left(a-c\right)}{\left(2x+2y-1\right)\left(x+y+3\right)}$

$=\frac{a\left(x-5y\right)}{\left(x-2y\right)\left(2x+2y-1\right)}$