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) $\left(\frac{1}{3}{y}^2-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2y^2\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^2+2\right)$
(v) $-\frac{1}{2}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{5}c{b}^2{a}^2+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b.$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)x^2-3x+5-\frac{1}{2}\left(3x^2-5x+7\right) \\ =x^2-3x+5-\frac{3x^2}{2}+\frac{5x}{2}-\frac{7}{2} \\ =x^2-\frac{3x^2}{2}-3x+\frac{5x}{2}+5-\frac{7}{2}(\mathrm{Collecting}\mathrm{like}\mathrm{terms}) \\ =\left(\frac{1-3}{2}\right)x^2+\left(\frac{-3+5}{2}\right)x+\left(\frac{10-7}{2}\right) \\ =-\frac{x^2}{2}-\frac{x}{2}+\frac{3}{2} \end{gathered}$$

Thus, the answer is $-\frac{x^2}{2}-\frac{x}{2}+\frac{3}{2}$.

$$\begin{gathered} \left(\mathrm{ii}\right)\left[5-3x+2y-\left(2x-y\right)\right]-\left(3x-7y+9\right) \\ =\left[5-3x+2y-2x+y\right]-\left(3x-7y+9\right) \\ =\left[5-5x+3y\right]-\left(3x-7y+9\right) \\ =5-5x+3y-3x+7y-9 \\ =5-9-5x-3x+3y+7y \\ =-4-8x+10y \end{gathered}$$

$\left(\mathrm{iii}\right)\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^2+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^2+\frac{1}{2}xy$
$=\frac{11}{2}{x}^{2}y+\frac{1}{15}y{x}^2-\frac{9}{4}x{y}^2-\frac{1}{14}{y}^{2}x+\frac{1}{4}xy+\frac{1}{2}xy$ (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)

$$\begin{gathered} \left(\mathrm{iv}\right)\left(\frac{1}{3}{y}^2-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2y^2\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^2+2\right) \\ =\frac{1}{3}{y}^2-\frac{4}{7}y+11-\frac{1}{7}y+3-2y^2-\frac{2}{7}y+\frac{2}{3}{y}^2-2 \end{gathered}$$

$=\frac{1}{3}{y}^2-2y^2+\frac{2}{3}{y}^2-\frac{4}{7}y-\frac{1}{7}y-\frac{2}{7}y+11+3-2$ (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)