Question

Simplify each of the following: $\left(i\right)x^2-3x+5-\frac{1}{2}(3x^2-5x+7)\left(ii\right)[5-3x+2y-(2x-y\left)\right]-(3x-7y+9)\left(iii\right)\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^2+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^2+\frac{1}{2}xy\left(iv\right)(\frac{1}{3}{y}^2-\frac{4}{7}y+11)-(\frac{1}{7}y-3+2y^2)-(\frac{2}{7}y-\frac{2}{3}{y}^2+2)\left(v\right)-\frac{1}{2}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{5}c{b}^2{a}^2+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b.$

Answer 1

$\left(i\right)x^2-3x+5-\frac{1}{2}(3x^2-5x+7)=x^2-3x+5-\frac{3}{2}{x}^2+\frac{5}{2}x-\frac{7}{2}=x^2-\frac{3}{2}{x}^2-3x+\frac{5}{2}x+5-\frac{7}{2}=(1-\frac{3}{2})x^2-(3-\frac{5}{2})x+(\frac{5}{1}-\frac{7}{2})=\frac{2-3}{2}{x}^2-\frac{6-5}{2}x+\frac{10-7}{2}=\frac{-1}{2}{x}^2-\frac{1}{2}x+\frac{3}{2}\left(ii\right)[5-3x+2y-(2x-y\left)\right]-(3x-7y+9)=[5-3x+2y-2x+y]-(3x-7y+9)=5-3x+2y-2x+y-3x+7y-9=5-9-3x-2x-3x+2y+y+7y=-4-8x+10y\left(iii\right)\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^2+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^2+\frac{1}{2}xy=\frac{11}{2}{x}^{2}y+\frac{1}{15}y{x}^2-\frac{9}{4}x{y}^2-\frac{1}{14}{y}^{2}x+\frac{1}{4}xy+\frac{1}{2}xy=(\frac{11}{2}+\frac{1}{15})x^{2}y-(\frac{9}{4}+\frac{1}{14})x{y}^2+(\frac{1}{4}+\frac{1}{2})xy=\frac{165+2}{30}{x}^{2}y-\frac{63+2}{28}x{y}^2+\frac{1+2}{4}xy=\frac{167}{30}{x}^{2}y-\frac{65}{28}x{y}^2+\frac{3}{4}xy\left(iv\right)(\frac{1}{3}{y}^2-\frac{4}{7}y+11)-(\frac{1}{7}y-3+2y^2)-(\frac{2}{7}y-\frac{2}{3}{y}^2+2)=\frac{1}{3}{y}^2-\frac{4}{7}y+11-\frac{1}{7}y+3-2y^2-\frac{2}{7}y+\frac{2}{3}y2-2=\frac{1}{3}{y}^2-2y^2+\frac{2}{3}{y}^2-\frac{4}{7}y-\frac{1}{7}y-\frac{2}{7}y+11+3-2=(\frac{1}{3}-\frac{2}{1}+\frac{2}{3})y^2-(\frac{4}{7}+\frac{1}{7}+\frac{2}{7})y+12=\frac{1-6+2}{3}{y}^2-\frac{4+1+2}{7}y+12=\frac{-3}{3}{y}^2-\frac{7}{7}y+12=-1y^2=1y+12=-y^2-y+12\left(v\right)-\frac{1}{2}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{5}c{b}^2{a}^2+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b=-\frac{1}{2}{a}^2{b}^{2}c-\frac{1}{5}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c+\frac{1}{6}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{7}ab{c}^2+\frac{1}{8}c{a}^{2}b=(-\frac{1}{2}-\frac{1}{5})a^2{b}^{2}c+(\frac{1}{3}+\frac{1}{6})a{b}^{2}c-(\frac{1}{4}+\frac{1}{7})ab{c}^2+\frac{1}{8}{a}^{2}bc=\frac{-5-2}{10}{a}^2{b}^{2}c+\frac{2+1}{6}a{b}^{2}c-\frac{7+4}{28}ab{c}^2+\frac{1}{8}{a}^{2}bc=\frac{-7}{10}{a}^2{b}^{2}c+\frac{3}{6}a{b}^{2}c-\frac{11}{28}ab{c}^2+\frac{1}{8}{a}^{2}bc=\frac{-7}{10}{a}^2{b}^{2}c+\frac{1}{2}a{b}^{2}c-\frac{11}{28}ab{c}^2+\frac{1}{8}{a}^{2}bc$