Simplify each of the following:
Question:

Simplify each of the following:

(i) $x^{2}-3 x+5-\frac{1}{2}\left(3 x^{2}-5 x+7\right)$

(ii) $[5-3 x+2 y-(2 x-y)]-(3 x-7 y+9)$

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

(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) $-\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+\frac{1}{6} c b^{2} a-\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b$.

Solution:

(i) $x^{2}-3 x+5-\frac{1}{2}\left(3 x^{2}-5 x+7\right)$

$=x^{2}-3 x+5-\frac{3 x^{2}}{2}+\frac{5 x}{2}-\frac{7}{2}$

$=x^{2}-\frac{3 x^{2}}{2}-3 x+\frac{5 x}{2}+5-\frac{7}{2}$            (Collecting like terms)

$=\left(\frac{1-3}{2}\right) x^{2}+\left(\frac{-3+5}{2}\right) x+\left(\frac{10-7}{2}\right)$

$=-\frac{x^{2}}{2}-\frac{x}{2}+\frac{3}{2}$

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

(ii) $[5-3 x+2 y-(2 x-y)]-(3 x-7 y+9)$

$=[5-3 x+2 y-2 x+y]-(3 x-7 y+9)$

$=[5-5 x+3 y]-(3 x-7 y+9)$

$=5-5 x+3 y-3 x+7 y-9$

$=5-9-5 x-3 x+3 y+7 y$

$=-4-8 x+10 y$

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

$=\frac{11}{2} x^{2} y+\frac{1}{15} y x^{2}-\frac{9}{4} x y^{2}-\frac{1}{14} y^{2} x+\frac{1}{4} x y+\frac{1}{2} x y \quad$ (Collecting like terms)

$=\left(\frac{165+2}{30}\right) x^{2} y+\left(\frac{-63-2}{28}\right) x y^{2}+\left(\frac{1+2}{4}\right) x y$

$=\frac{167}{30} x^{2} y-\frac{65}{28} y^{2} x+\frac{3}{2} x y$          (Collecting like terms)

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

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

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

$=\left(\frac{1-6+2}{3}\right) y^{2}+\left(\frac{-4-1-2}{7}\right) y+12$

$=-y^{2}-7 y+12$        (Combining like terms)

(v) $-\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+\frac{1}{6} c b^{2} a-\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b$

$=-\frac{1}{2} a^{2} b^{2} c-\frac{1}{5} c b^{2} a^{2}+\frac{1}{3} a b^{2} c+\frac{1}{6} c b^{2} a-\frac{1}{4} a b c^{2}-\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b$     (Combining like terms)  

$=\left(\frac{-5-2}{10}\right) a^{2} b^{2} c+\left(\frac{2+1}{6}\right) c b^{2} a^{2}+\left(\frac{-7-4}{28}\right) c^{2} a b+\frac{1}{8} c a^{2} b$

 

$=-\frac{7}{10} a^{2} b^{2} c+\frac{1}{2} a b^{2} c-\frac{11}{28} a b c^{2}+\frac{1}{8} a^{2} b c$           (Combining like terms)

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