Take away:(i) \( \frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x \) from \( \frac{x^{3}}{3}-\frac{5}{2} x^{2}+ \) \( \frac{3}{5} x+\frac{1}{4} \)(ii) \( \frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}-\frac{6}{5} \) from \( \frac{1}{3} a^{3}-\frac{3}{4} a^{2}- \) \( \frac{5}{2} \)(iii) \( \frac{7}{4} x^{3}+\frac{3}{5} x^{2}+\frac{1}{2} x+\frac{9}{2} \) from \( \frac{7}{2}-\frac{x}{3}- \) \( \frac{x^{2}}{5} \)(iv) \( \frac{y^{3}}{3}+\frac{7}{3} y^{2}+\frac{1}{2} y+\frac{1}{2} \) from \( \frac{1}{3}-\frac{5}{3} y^{2} \)(v) \( \frac{2}{3} a c-\frac{5}{7} a b+\frac{2}{3} b c \) from \( \frac{3}{2} a b-\frac{7}{4} a c- \) \( \frac{5}{6} b c \)
To do:
We have to take away the given algebraic expressions.
Solution:
(i) $(\frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4})-(\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x)=\frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4}-\frac{6}{5} x^{2}+\frac{4}{5} x^{3}-\frac{5}{6}-\frac{3}{2} x$
$=\frac{x^{3}}{3}+\frac{4}{5} x^{3}-\frac{5}{2} x^{2}-\frac{6}{5} x^{2}+\frac{3}{5} x-\frac{3}{2} x+\frac{1}{4}-\frac{5}{6}$
$=(\frac{1}{3}+\frac{4}{5}) x^{3}+(\frac{-5}{2}-\frac{6}{5}) x^{2}+(\frac{3}{5}-\frac{3}{2}) x+(\frac{1}{4}-\frac{5}{6})$
$=\frac{5+12}{15} x^{3}+\frac{-25-12}{10} x^{2}+\frac{6-15}{10} x+\frac{3-10}{12}$
$=\frac{17}{15} x^{3}-\frac{37}{10} x^{2}-\frac{9}{10} x-\frac{7}{12}$
(ii) $(\frac{1}{3} a^{3}-\frac{3}{4} a^{2}-\frac{5}{2})-(\frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}-\frac{6}{5})=\frac{1}{3} a^{3}-\frac{3}{4} a^{2}-\frac{5}{2}-\frac{5}{2} a^{2}-\frac{3}{2} a^{3}-\frac{a}{3}+\frac{6}{5}$
$=\frac{1}{3} a^{3}-\frac{3}{2} a^{3}-\frac{3}{4} a^{2}-\frac{5}{2} a^{2}-\frac{a}{3}+\frac{6}{5}-\frac{5}{2}$
$=(\frac{1}{3}-\frac{3}{2}) a^{3}-(\frac{3}{4}+\frac{5}{2}) a^{2}-(\frac{1}{3}) a+(\frac{6}{5}-\frac{5}{2})$
$=\frac{2-9}{6} a^{3}-\frac{3+10}{4} a^{2}-\frac{1}{3} a+\frac{12-25}{10}$
$=\frac{-7}{6} a^{3}-\frac{13}{4} a^{2}-\frac{1}{3} a-\frac{13}{10}$
(iii) $(\frac{7}{2}-\frac{x}{3}-\frac{x^{2}}{5})-(\frac{7}{4} x^{3}+\frac{3}{5} x^{2}+\frac{1}{2} x+\frac{9}{2})=\frac{7}{2}-\frac{x}{3}-\frac{x^{2}}{5}-\frac{7}{4} x^{3}-\frac{3}{5} x^{2}-\frac{1}{2} x-\frac{9}{2}$
$=\frac{-7}{4} x^{3}-\frac{x^{2}}{5}-\frac{3}{5} x^{2}-\frac{x}{3}-\frac{1}{2} x+\frac{7}{2}-\frac{9}{2}$
$=\frac{-7}{4} x^{3}-(\frac{1}{5}+\frac{3}{5}) x^{2}-(\frac{1}{3}+\frac{1}{2}) x+\frac{7-9}{2}$
$=\frac{-7}{4} x^{3}-\frac{1+3}{5} x^{2}-\frac{2+3}{6} x-\frac{2}{2}$
$=\frac{-7}{4} x^{3}-\frac{4}{5} x^{2}-\frac{5}{6} x-1$
(iv) $(\frac{1}{3}-\frac{5}{3} y^{2})-(\frac{y^{3}}{3}+\frac{7}{3} y^{2}+\frac{1}{2} y+\frac{1}{2})=\frac{1}{3}-\frac{5}{3} y^{2}-\frac{y^{3}}{3}-\frac{7}{3} y^{2}-\frac{1}{2} y-\frac{1}{2}$
$=\frac{-y^{3}}{3}-(\frac{5}{3}+\frac{7}{3}) y^{2}-\frac{1}{2} y-\frac{1}{2}+\frac{1}{3}$
$=\frac{-y^{3}}{3}-\frac{5+7}{3} y^{2}-\frac{1}{2} y-\frac{3-2}{6}$
$=\frac{-y^{3}}{3}-\frac{12}{3} y^{2}-\frac{1}{2} y-\frac{1}{6}$
$=\frac{-1}{3} y^{3}-4 y^{2}-\frac{1}{2} y-\frac{1}{6}$
(v) $(\frac{3}{2} a b-\frac{7}{4} a c-\frac{5}{6} b c)-(\frac{2}{3} a c-\frac{5}{7} a b+\frac{2}{3} b c)=(\frac{3}{2} a b-\frac{7}{4} a c-\frac{5}{6} b c-\frac{2}{3} a c+\frac{5}{7} a b-\frac{2}{3} b c)$
$=(\frac{3}{2} a b+\frac{5}{7} a b-\frac{5}{6} b c-\frac{2}{3} b c-\frac{7}{4} a c-\frac{2}{3} a c)$
$=(\frac{3}{2}+\frac{5}{7}) a b-(\frac{5}{6}+\frac{2}{3}) b c-(\frac{7}{4}+\frac{2}{3}) a c$
$=\frac{21+10}{14} a b-\frac{5+4}{6} b c-\frac{21+8}{12} a c$
$=\frac{31}{14} a b-\frac{9}{6} b c-\frac{29}{12} a c$
$=\frac{31}{14} a b-\frac{3}{2} b c-\frac{29}{12} a c$
Related Articles Add the following algebraic expressions(i) \( 3 a^{2} b,-4 a^{2} b, 9 a^{2} b \)(ii) \( \frac{2}{3} a, \frac{3}{5} a,-\frac{6}{5} a \)(iii) \( 4 x y^{2}-7 x^{2} y, 12 x^{2} y-6 x y^{2},-3 x^{2} y+5 x y^{2} \)(iv) \( \frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+ \) \( \frac{5}{2} b-\frac{5}{4} c \)(v) \( \frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y \)(vi) \( \frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3} \) \( \frac{3}{2} x^{2}-\frac{5}{2} x-2 \) Subtract:(i) $-5xy$ from $12xy$(ii) $2a^2$ from $-7a^2$(iii) \( 2 a-b \) from \( 3 a-5 b \)(iv) \( 2 x^{3}-4 x^{2}+3 x+5 \) from \( 4 x^{3}+x^{2}+x+6 \)(v) \( \frac{2}{3} y^{3}-\frac{2}{7} y^{2}-5 \) from \( \frac{1}{3} y^{3}+\frac{5}{7} y^{2}+y-2 \)(vi) \( \frac{3}{2} x-\frac{5}{4} y-\frac{7}{2} z \) from \( \frac{2}{3} x+\frac{3}{2} y-\frac{4}{3} z \)(vii) \( x^{2} y-\frac{4}{5} x y^{2}+\frac{4}{3} x y \) from \( \frac{2}{3} x^{2} y+\frac{3}{2} x y^{2}- \) \( \frac{1}{3} x y \)(viii) \( \frac{a b}{7}-\frac{35}{3} b c+\frac{6}{5} a c \) from \( \frac{3}{5} b c-\frac{4}{5} a c \) Take away:\( \frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x \) from \( \frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4} \) \Find $(x +y) \div (x - y)$. if,(i) \( x=\frac{2}{3}, y=\frac{3}{2} \)(ii) \( x=\frac{2}{5}, y=\frac{1}{2} \)(iii) \( x=\frac{5}{4}, y=\frac{-1}{3} \)(iv) \( x=\frac{2}{7}, y=\frac{4}{3} \)(v) \( x=\frac{1}{4}, y=\frac{3}{2} \) Solve(a) \( \frac{2}{3}+\frac{1}{7} \)(b) \( \frac{3}{10}+\frac{7}{15} \)(c) \( \frac{4}{9}+\frac{2}{7} \)(d) \( \frac{5}{7}+\frac{1}{3} \)(e) \( \frac{2}{5}+\frac{1}{6} \)(f) \( \frac{4}{5}+\frac{2}{3} \)(g) \( \frac{3}{4}-\frac{1}{3} \)(h) \( \frac{5}{6}-\frac{1}{3} \)(i) \( \frac{2}{3}+\frac{3}{4}+\frac{1}{2} \)(j) \( \frac{1}{2}+\frac{1}{3}+\frac{1}{6} \)(k) \( 1 \frac{1}{3}+3 \frac{2}{3} \)(l) \( 4 \frac{2}{3}+3 \frac{1}{4} \)(m) \( \frac{16}{5}-\frac{7}{5} \)(n) \( \frac{4}{3}-\frac{1}{2} \) Simplify:(i) \( \frac{-3}{2}+\frac{5}{4}-\frac{7}{4} \)(ii) \( \frac{5}{3}-\frac{7}{6}+\frac{-2}{3} \)(iii) \( \frac{5}{4}-\frac{7}{6}-\frac{-2}{3} \)(iv) \( \frac{-2}{5}-\frac{-3}{10}-\frac{-4}{7} \)(v) \( \frac{5}{6}+\frac{-2}{5}-\frac{-2}{15} \)(vi) \( \frac{3}{8}-\frac{-2}{9}+\frac{-5}{36} \) Solve:$\frac{3 x}{2}-\frac{5 y}{3}=-2$$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$ (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) \( -\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+ \) \( \frac{1}{6} c b^{2} a+\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b \). Solve the following pairs of equations by reducing them to a pair of linear equations:(i) \( \frac{1}{2 x}+\frac{1}{3 y}=2 \)\( \frac{1}{3 x}+\frac{1}{2 y}=\frac{13}{6} \)(ii) \( \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \)\( \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \)(iii) \( \frac{4}{x}+3 y=14 \)\( \frac{3}{x}-4 y=23 \)(iv) \( \frac{5}{x-1}+\frac{1}{y-2}=2 \)\( \frac{6}{x-1}-\frac{3}{y-2}=1 \)(v) \( \frac{7 x-2 y}{x y}=5 \)\( \frac{8 x+7 y}{x y}=15 \),b>(vi) \( 6 x+3 y=6 x y \)\( 2 x+4 y=5 x y \)4(vii) \( \frac{10}{x+y}+\frac{2}{x-y}=4 \)\( \frac{15}{x+y}-\frac{5}{x-y}=-2 \)(viii) \( \frac{1}{3 x+y}+\frac{1}{3 x-y}=\frac{3}{4} \)\( \frac{1}{2(3 x+y)}-\frac{1}{2(3 x-y)}=\frac{-1}{8} \). Solve:(i) $3-\frac{2}{5}$(ii) $4+\frac{7}{8}$(iii) $\frac{3}{5}+\frac{2}{7}$(iv) $\frac{9}{11}-\frac{4}{15}$(v) $\frac{7}{10}+\frac{2}{5}+\frac{3}{2}$(vi) $2\frac{2}{3}+3\frac{1}{2}$(vii) $8\frac{1}{2}-3\frac{5}{8}$ Find:(i) $\frac{2}{5}\div\frac{1}{2}$(ii) $\frac{4}{9}\div\frac{2}{3}$(iii) $\frac{3}{7}\div\frac{8}{7}$(iv) $2\frac{1}{3}\div\frac{3}{5}$(v) $3\frac{1}{2}\div\frac{8}{3}$(vi) $\frac{2}{5}\div1\frac{1}{2}$(vii) $3\frac{1}{5}\div1\frac{2}{3}$(viii) $2\frac{1}{5}\div1\frac{1}{5}$ Multiply the following fractions:(i) $\frac{2}{5}\times5\frac{1}{4}$(ii) $6\frac{2}{5}\times\frac{7}{9}$(iii) $\frac{3}{2}\times5\frac{1}{3}$(iv) $\frac{5}{6}\times2\frac{3}{7}$(v) $3\frac{2}{5}\times\frac{4}{7}$(vi) $2\frac{3}{5}\times3$(vii) $3\frac{4}{7}\times\frac{3}{5}$ Verify associativity of addition of rational numbers i.e., $(x + y) + z = x + (y + z)$, when:(i) \( x=\frac{1}{2}, y=\frac{2}{3}, z=-\frac{1}{5} \)(ii) \( x=\frac{-2}{5}, y=\frac{4}{3}, z=\frac{-7}{10} \)(iii) \( x=\frac{-7}{11}, y=\frac{2}{-5}, z=\frac{-3}{22} \)(iv) \( x=-2, y=\frac{3}{5}, z=\frac{-4}{3} \) Solve:\( \frac{3 x}{5}+4+x-2=\frac{\frac{3 x}{5} \times x}{2} \) Verify:$\frac{-2}{5} + [\frac{3}{5} + \frac{1}{2}] = [\frac{-2}{5} + \frac{3}{5}] + \frac{1}{2}$
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