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Question

# Simplify each of the following: (i) x2 − 3x + 5 − $\frac{1}{2}$ (3x2 − 5x + 7) (ii) [5 − 3x + 2y − (2x − y)] − (3x − 7y + 9) (iii) $\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^{2}+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^{2}+\frac{1}{2}xy$ (iv) $\left(\frac{1}{3}{y}^{2}-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2{y}^{2}\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^{2}+2\right)$ (v) $-\frac{1}{2}{a}^{2}{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^{2}-\frac{1}{5}c{b}^{2}{a}^{2}+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b.$

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Solution

## $\left(\mathrm{i}\right){x}^{2}-3x+5-\frac{1}{2}\left(3{x}^{2}-5x+7\right)\phantom{\rule{0ex}{0ex}}={x}^{2}-3x+5-\frac{3{x}^{2}}{2}+\frac{5x}{2}-\frac{7}{2}\phantom{\rule{0ex}{0ex}}={x}^{2}-\frac{3{x}^{2}}{2}-3x+\frac{5x}{2}+5-\frac{7}{2}\left(\mathrm{Collecting}\mathrm{like}\mathrm{terms}\right)\phantom{\rule{0ex}{0ex}}=\left(\frac{1-3}{2}\right){x}^{2}+\left(\frac{-3+5}{2}\right)x+\left(\frac{10-7}{2}\right)\phantom{\rule{0ex}{0ex}}=-\frac{{x}^{2}}{2}-\frac{x}{2}+\frac{3}{2}$ Thus, the answer is $-\frac{{x}^{2}}{2}-\frac{x}{2}+\frac{3}{2}$. $\left(\mathrm{ii}\right)\left[5-3x+2y-\left(2x-y\right)\right]-\left(3x-7y+9\right)\phantom{\rule{0ex}{0ex}}=\left[5-3x+2y-2x+y\right]-\left(3x-7y+9\right)\phantom{\rule{0ex}{0ex}}=\left[5-5x+3y\right]-\left(3x-7y+9\right)\phantom{\rule{0ex}{0ex}}=5-5x+3y-3x+7y-9\phantom{\rule{0ex}{0ex}}=5-9-5x-3x+3y+7y\phantom{\rule{0ex}{0ex}}=-4-8x+10y$ $\left(\mathrm{iii}\right)\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^{2}+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^{2}+\frac{1}{2}xy$ $=\frac{11}{2}{x}^{2}y+\frac{1}{15}y{x}^{2}-\frac{9}{4}x{y}^{2}-\frac{1}{14}{y}^{2}x+\frac{1}{4}xy+\frac{1}{2}xy$ (Collecting like terms) =$\left(\frac{165+2}{30}\right){x}^{2}y+\left(\frac{-63-2}{28}\right)x{y}^{2}+\left(\frac{1+2}{4}\right)xy$ $=\frac{167}{30}{x}^{2}y-\frac{65}{28}{y}^{2}x+\frac{3}{2}xy$ (Combining like terms) $\left(\mathrm{iv}\right)\left(\frac{1}{3}{y}^{2}-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2{y}^{2}\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^{2}+2\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}{y}^{2}-\frac{4}{7}y+11-\frac{1}{7}y+3-2{y}^{2}-\frac{2}{7}y+\frac{2}{3}{y}^{2}-2$ $=\frac{1}{3}{y}^{2}-2{y}^{2}+\frac{2}{3}{y}^{2}-\frac{4}{7}y-\frac{1}{7}y-\frac{2}{7}y+11+3-2$ (Collecting like terms) =$\left(\frac{1-6+2}{3}\right){y}^{2}+\left(\frac{-4-1-2}{7}\right)y+12$ $=-{y}^{2}-7y+12$ (Combining like terms) $\left(\mathrm{v}\right)-\frac{1}{2}{a}^{2}{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^{2}-\frac{1}{5}c{b}^{2}{a}^{2}+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b$ $=-\frac{1}{2}{a}^{2}{b}^{2}c-\frac{1}{5}c{b}^{2}{a}^{2}+\frac{1}{3}a{b}^{2}c+\frac{1}{6}c{b}^{2}a-\frac{1}{4}ab{c}^{2}-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b$ (Collecting like terms) =$\left(\frac{-5-2}{10}\right){a}^{2}{b}^{2}c+\left(\frac{2+1}{6}\right)c{b}^{2}{a}^{2}+\left(\frac{-7-4}{28}\right){c}^{2}ab+\frac{1}{8}c{a}^{2}b$ $=-\frac{7}{10}{a}^{2}{b}^{2}c+\frac{1}{2}a{b}^{2}c-\frac{11}{28}ab{c}^{2}+\frac{1}{8}{a}^{2}bc$ (Combining like terms)

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