(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) \( -\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+ \) \( \frac{1}{6} c b^{2} a+\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b \).
To do:
We have to solve the given expressions.
Solution:
(i) $x^{2}-3 x+5-\frac{1}{2}(3 x^{2}-5 x+7) =x^{2}-3 x+5-\frac{3}{2} x^{2}+\frac{5}{2} x-\frac{7}{2}$
$=x^{2}-\frac{3}{2} x^{2}-3 x+\frac{5}{2} x+5-\frac{7}{2}$
$=(1-\frac{3}{2}) x^{2}-(3-\frac{5}{2}) x+(\frac{5}{1}-\frac{7}{2})$
$=\frac{2-3}{2} x^{2}-\frac{6-5}{2} x+\frac{10-7}{2}$
$=\frac{-1}{2} x^{2}-\frac{1}{2} x+\frac{3}{2}$
(ii) $[5-3 x+2 y-(2 x-y)]-(3 x-7 y+9)=[5-3 x+2 y-2 x+y]-(3 x-7 y+9)$
$=5-3 x+2 y-2 x+y-3 x+7 y-9$
$=5-9-3 x-2 x-3 x+2 y+y+7 y$
$=-4-8 x+10 y$
(iii) $\frac{11}{2} x^{2} y-\frac{9}{4} x y^{2}+\frac{1}{4} x y-\frac{1}{14} y^{2} x+\frac{1}{15} y x^{2}+\frac{1}{2} x y=\frac{11}{2} x^{2} y+\frac{1}{15} y x^{2}-\frac{9}{4} x y^{2}-\frac{1}{14} y^{2} x+\frac{1}{4} x y+\frac{1}{2} x y$
$=(\frac{165+2}{30}) x^{2} y+(\frac{-63-2}{28}) x y^{2}+(\frac{1+2}{4}) x y$
$=\frac{167}{30} x^{2} y-\frac{65}{28} y^{2} x+\frac{3}{2} x y$
(iv) $(\frac{1}{3} y^{2}-\frac{4}{7} y+11)-(\frac{1}{7} y-3+2 y^{2})-(\frac{2}{7} y-\frac{2}{3} y^{2}+2)=\frac{1}{3} y^{2}-\frac{4}{7} y+11-\frac{1}{7} y+3-2 y^{2}-\frac{2}{7} y+\frac{2}{3} y^{2}-2$
$=\frac{1}{3} y^{2}-2 y^{2}+\frac{2}{3} y^{2}-\frac{4}{7} y-\frac{1}{7} y-\frac{2}{7} y+11+3-2$
$=(\frac{1-6+2}{3}) y^{2}+(\frac{-4-1-2}{7}) y+12$
$=-y^{2}-7 y+12$
(v) $-\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+\frac{1}{6} c b^{2} a-\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b=-\frac{1}{2} a^{2} b^{2} c-\frac{1}{5} a^{2} b^{2} c+\frac{1}{3} a b^{2} c+\frac{1}{6} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{7} a b c^{2}+\frac{1}{8} c a^{2} b$
$=(\frac{1}{2}-\frac{1}{5}) a^{2} b^{2} c+(\frac{1}{3}+\frac{1}{6}) a b^{2} c-(\frac{1}{4}+\frac{1}{7}) a b c^2+\frac{1}{8} a^{2} b c$
$=\frac{-5-2}{10} a^{2} b^{2} c+\frac{2+1}{6} a b^{2} c-\frac{7+4}{28} a b c^{2}+\frac{1}{8} a^{2} b c$
$=\frac{-7}{10} a^{2} b^{2} c+\frac{3}{6} a b^{2} c-\frac{11}{28} a b c^{2}+\frac{1}{8} a^{2} b c$
$=\frac{-7}{10} a^{2} b^{2} c+\frac{1}{2} a b^{2} c-\frac{11}{28} a b c^{2}+\frac{1}{8} a^{2} b c$
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