Question

Solve :

$\left(i\right)\frac{1}{3}x-6=\frac{5}{2}\left(ii\right)\frac{2x}{3}-\frac{3x}{8}=\frac{7}{12}\left(iii\right)(x+2)(x+3)+(x-3)(x-2)-2x(x+1)=0\left(iv\right)\frac{1}{10}-\frac{7}{x}=35\left(v\right)13(x-4)-3(x-9)-4(x+4)=0\left(vi\right)x+7-\frac{8x}{3}=\frac{17x}{6}-\frac{5x}{8}\left(vii\right)\frac{3x-2}{4}-\frac{2x+3}{3}=\frac{2}{3}-x\left(viii\right)\frac{x+2}{6}-(\frac{11-x}{3}-\frac{1}{4})=\frac{3x-4}{12}\left(ix\right)\frac{2}{5x}-\frac{5}{3x}=\frac{1}{15}\left(x\right)\frac{x+2}{3}-\frac{x+1}{5}=\frac{x-3}{4}-1\left(xi\right)\frac{3x-2}{3}+\frac{2x+3}{2}=x+\frac{7}{6}\left(xii\right)x-\frac{x-1}{2}=1-\frac{x-2}{3}\left(xiii\right)\frac{9x+7}{2}-(x-\frac{x-2}{7})=36\left(xiv\right)\frac{6x+1}{2}+1=\frac{7x-3}{3}$

Answer 1

$\left(i\right)\frac{1}{3}x-6=\frac{5}{2}\Rightarrow \frac{1}{3}x=\frac{5}{2}+\frac{6}{1}\Rightarrow \frac{1}{3}x=\frac{5\times 1}{2\times 1}+\frac{6\times 2}{1\times 2}\Rightarrow \frac{1}{3}x=\frac{5}{2}+\frac{12}{2}\Rightarrow \frac{1}{3}x=\frac{5+12}{2}=\frac{1}{3}x=\frac{17}{2}.=x=\frac{17\times 3}{2\times 1}=\frac{51}{2}=25\frac{1}{2}\left(ii\right)\frac{2x}{3}-\frac{3x}{8}=\frac{7}{12}\begin{array}{cc}2& 3,8\\ 2& 3,4\\ 2& 3,2\\ 3& 3,1\\ & 1\end{array}\text{L.C.M. of 3 and 8 =}2\times 2\times 2\times 3=24\therefore \frac{2x\times 8}{3\times 8}-\frac{3x\times 8}{8\times 3}=\frac{7}{12}=\frac{16x}{24}-\frac{9x}{24}=\frac{7}{12}=\frac{16x-9x}{24}=\frac{7}{12}=\frac{7x}{24}=\frac{7}{12}=x=\frac{7\times 27}{12\times 7}=2\therefore x=2\left(iii\right)(x+2)(x+3)+(x-3)(x-2)-2x(x+1)=0\Rightarrow [x^2+(2+3)x+2\times 3]+[x^2+(-3-2)x+(-3\left)\right(-2\left)\right]-2x^2-2x=0\Rightarrow x^2+5x+6+x^2-5x+6-2x^2-2x=0\Rightarrow x^2+x^2-2x^2+5x-5x-2x+6+6=0=-2x+12=0\text{Subtracting 12 from both sides,}-2x+12-12=0-12\Rightarrow -2x=-12\text{Dividing by -2}\frac{-2x}{-2}=\frac{-12}{-2}\Rightarrow x=6\therefore x=6\text{Verification}\text{L.H.S.}=(x+2)(x+3)+(x-3)(x-2)-2x(x+1)=(6+2)(6+3)+(6-3)(6-2)-2\times 6(6+1)=8\times 9+3\times 4-12\times 7=72+12-84=84-84=0=\text{R.H.S.}\left(iv\right)\frac{1}{10}-\frac{7}{x}=35\Rightarrow \frac{-7}{x}=35-\frac{1}{10}\Rightarrow \frac{-7}{x}=\frac{35\times 10}{1\times 10}-\frac{1\times 1}{10\times 1}\Rightarrow \frac{-7}{x}=\frac{350-1}{10}\Rightarrow \frac{1}{x}=\frac{350-1}{10\times (-7)}\Rightarrow x=\frac{349}{(-70)}=\frac{-70}{349}\left(v\right)13(x-4)-3(x-9)-4(x+4)=0\Rightarrow 13(x-4)-3(x-9)-5(x+4)=0\Rightarrow 13x-52-3x+27-5x-20=0\Rightarrow 13x-3x-5x-52+27-20=0\Rightarrow 13x-8x-72+27=0\Rightarrow 5x-45=0\text{Dividing by 5,}\frac{5x}{5}-\frac{45}{5}=0\Rightarrow x-9=0\Rightarrow x=9\text{Verification,~}\text{L.H.S.}=13(x-4)-3(x-9)-5(x+4)=13(9-4)-3(9-9)-5(9+4)=13\times 5-3\times 0-5\times 13\left(vi\right)x+7-\frac{8x}{3}=\frac{17x}{6}-\frac{5x}{8}\Rightarrow \frac{3(x+7)-8x}{3}=\frac{17x\times 4-5x\times 3}{24}\Rightarrow \frac{3x+21-8x}{3}=\frac{68x-15x}{24}\Rightarrow \frac{-5x+21}{3}=\frac{53x}{24}\Rightarrow 3\times 53x=24(-5x+21)\Rightarrow 159x=-120x+504\Rightarrow 159x+120x=504\Rightarrow 279x=504\Rightarrow x=\frac{504}{279}=\frac{168}{93}=\frac{56}{61}\therefore x=1\frac{25}{31}\left(vii\right)\frac{3x-2}{4}-\frac{2x+3}{3}=\frac{2}{3}-x=\frac{3x-2}{4}-\frac{2x+3}{3}=\frac{2}{3}-\frac{x}{1}=\frac{3(3x-2)-4(2x+3)}{12}=\frac{2\times 1}{3\times 1}-\frac{x\times 3}{1\times 3}=\frac{9x-6-8x-12}{12}=\frac{2-3x}{3}=\frac{(x-18)}{12}=\frac{2-3x}{3}=3(x-18)=12(2-3x)=3x-54=24-36x=3x+36x=24+54=39x=78x=\frac{78}{39}=2\therefore x=2\left(viii\right)\frac{x+2}{6}-(\frac{11-x}{3}-\frac{1}{4})=\frac{3x-4}{12}\Rightarrow \frac{x+2}{6}-\left(\frac{4(11-x)-1\times 3}{12}\right)=\frac{3x-4}{12}\Rightarrow \frac{x+2}{6}-\frac{44+4x+3}{12}=\frac{3x-4}{12}\Rightarrow \frac{2(x+2)-41+4x}{12}=\frac{3x-4}{12}\Rightarrow \frac{2x+4-41+4x}{12}=\frac{3x-4}{12}\Rightarrow \frac{6x-37}{12}=\frac{3x-4}{12}\Rightarrow 12(6x-37)=12(3x-4)\Rightarrow 72x-444=36x-48\Rightarrow 72x-36x=-48+444\Rightarrow 36x=396\Rightarrow x=\frac{396}{36}=11\therefore x=11\left(ix\right)\frac{2}{5x}-\frac{5}{3x}=\frac{1}{15}\frac{2}{5x}-\frac{5}{3x}=\frac{1}{15}\Rightarrow \frac{2\times 3}{5x\times 3}-\frac{5\times 5}{3x\times 5}=\frac{1}{15}\Rightarrow \frac{6-25}{15x}=\frac{1}{15}\Rightarrow \frac{-19}{15x}=\frac{1}{15}\Rightarrow \frac{-19}{x}=\frac{15}{15}\Rightarrow -19=x\therefore x=-19\left(x\right)\frac{x+2}{3}-\frac{x+1}{5}=\frac{x-3}{4}-1\text{(L.C.M. of 3 and 5 = 15)}\Rightarrow \frac{5(x+2)-3(x+1)}{15}=\frac{x-3-4}{4}\Rightarrow \frac{5x+10-3x-3}{15}=\frac{x-7}{4}\Rightarrow \frac{2x+7}{15}=\frac{x-7}{4}\Rightarrow 4(2x+7)=15(x-7)\Rightarrow 8x+28=15x-105\Rightarrow 8x-15x=-133x=\frac{-133}{-7}\therefore x=19\left(xi\right)\frac{3x-2}{3}+\frac{2x+3}{2}=x+\frac{7}{6}\Rightarrow \frac{2(3x-2)+3(2x+3)}{6}=x+\frac{7}{6}\Rightarrow \frac{6x-4+6x+9}{6}=\frac{6x+7}{6}\Rightarrow \frac{12x+5}{6}=\frac{6x+7}{6}\Rightarrow 6(12x+5)=6(6x+7)\Rightarrow 72x+30=36x-42\Rightarrow 72x-36x=42-30\Rightarrow 36x=12x=\frac{12}{36}\therefore x=\frac{1}{3}\left(xii\right)x-\frac{x-1}{2}=1-\frac{x-2}{3}\Rightarrow \frac{2\left(x\right)-1(x-1)}{2}=\frac{3\left(1\right)-1(x-2)}{3}\Rightarrow \frac{2x-x+1}{2}=\frac{3-x+2}{3}\Rightarrow \frac{1x+1}{2}=\frac{5-x}{3}\Rightarrow 3(x+1)=2(5-x)\Rightarrow 3x+3=10-2x\Rightarrow 3x+2x=10-3\Rightarrow 5x=7\therefore x=\frac{7}{5}\left(xiii\right)\frac{9x+7}{2}-(x-\frac{x-2}{7})=36\Rightarrow \frac{9x+7}{2}-\left(\frac{7\times x-1(x-2)}{7}\right)=36\Rightarrow \frac{9x+7}{2}-\left(\frac{7x-x-2}{7}\right)=36\Rightarrow \frac{9x+7}{2}-\left(\frac{6x-2}{7}\right)=36\Rightarrow \frac{7(9x+7)+2(-6x+2)}{14}=36\Rightarrow \frac{63x+49-12x+4}{14}=36\Rightarrow \frac{51x+53}{14}=36\Rightarrow 51x+53=14\times 36\Rightarrow 51x=504-53\Rightarrow 51x=459\Rightarrow x=\frac{459}{51}\therefore x=9\left(xiv\right)\frac{6x+1}{2}+1=\frac{7x-3}{3}\Rightarrow \frac{(6x+1)}{2}+1=\frac{7x-3}{3}\Rightarrow \frac{6x+1+2}{2}=\frac{7x-3}{3}\Rightarrow \frac{6x+3}{2}=\frac{7x-3}{3}\Rightarrow 3(6x+3)=2(7x-3)\Rightarrow 18x+9=14x-6\Rightarrow 18x-14x=-6-9\Rightarrow 4x=-15\therefore x=\frac{-15}{4}$