Solve 53/33:(4/11+2/3-frac{5/22)}{sqrt{(frac{7}{8}+frac{2}{5}+frac{13}{40})*frac{1}{10}}}:(1+frac{7}{5})=sqrt{frac{125}{6}*(frac{4}{3}-frac{5}{8}+frac{3}{24})}:x

$$\frac { \frac { 53 } { 33 } : ( \frac { 4 } { 11 } + \frac { 2 } { 3 } - \frac { 5 } { 22 } ) } { \sqrt { ( \frac { 7 } { 8 } + \frac { 2 } { 5 } + \frac { 13 } { 40 } ) \cdot \frac { 1 } { 10 } } } : ( 1 + \frac { 7 } { 5 } ) = \sqrt { \frac { 125 } { 6 } \cdot ( \frac { 4 } { 3 } - \frac { 5 } { 8 } + \frac { 3 } { 24 } ) } : x$$

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{4}{11}+\frac{2}{3}-\frac{5}{22}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Least common multiple of 11 and 3 is 33. Convert \frac{4}{11} and \frac{2}{3} to fractions with denominator 33.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{12}{33}+\frac{22}{33}-\frac{5}{22}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Since \frac{12}{33} and \frac{22}{33} have the same denominator, add them by adding their numerators.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{12+22}{33}-\frac{5}{22}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Add 12 and 22 to get 34.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{34}{33}-\frac{5}{22}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Least common multiple of 33 and 22 is 66. Convert \frac{34}{33} and \frac{5}{22} to fractions with denominator 66.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{68}{66}-\frac{15}{66}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Since \frac{68}{66} and \frac{15}{66} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{68-15}{66}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Subtract 15 from 68 to get 53.

$$\frac{5}{12}x\times \frac{\frac{\frac{53}{33}}{\frac{53}{66}}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Divide \frac{53}{33} by \frac{53}{66} by multiplying \frac{53}{33} by the reciprocal of \frac{53}{66}.

$$\frac{5}{12}x\times \frac{\frac{53}{33}\times \frac{66}{53}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Multiply \frac{53}{33} times \frac{66}{53} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{12}x\times \frac{\frac{53\times 66}{33\times 53}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Cancel out 53 in both numerator and denominator.

$$\frac{5}{12}x\times \frac{\frac{66}{33}}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Divide 66 by 33 to get 2.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\left(\frac{7}{8}+\frac{2}{5}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Least common multiple of 8 and 5 is 40. Convert \frac{7}{8} and \frac{2}{5} to fractions with denominator 40.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\left(\frac{35}{40}+\frac{16}{40}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Since \frac{35}{40} and \frac{16}{40} have the same denominator, add them by adding their numerators.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\left(\frac{35+16}{40}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Add 35 and 16 to get 51.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\left(\frac{51}{40}+\frac{13}{40}\right)\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Since \frac{51}{40} and \frac{13}{40} have the same denominator, add them by adding their numerators.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{51+13}{40}\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Add 51 and 13 to get 64.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{64}{40}\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Reduce the fraction \frac{64}{40} to lowest terms by extracting and canceling out 8.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{8}{5}\times \frac{1}{10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Multiply \frac{8}{5} times \frac{1}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{8\times 1}{5\times 10}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Do the multiplications in the fraction \frac{8\times 1}{5\times 10}.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{8}{50}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Reduce the fraction \frac{8}{50} to lowest terms by extracting and canceling out 2.

$$\frac{5}{12}x\times \frac{2}{\sqrt{\frac{4}{25}}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Rewrite the square root of the division \frac{4}{25} as the division of square roots \frac{\sqrt{4}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{5}{12}x\times \frac{2}{\frac{2}{5}}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Divide 2 by \frac{2}{5} by multiplying 2 by the reciprocal of \frac{2}{5}.

$$\frac{5}{12}x\times 2\times \frac{5}{2}=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Cancel out 2 and 2.

$$\frac{5}{12}x\times 5=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Express \frac{5}{12}\times 5 as a single fraction.

$$\frac{5\times 5}{12}x=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Multiply 5 and 5 to get 25.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{4}{3}-\frac{5}{8}+\frac{3}{24}\right)}$$

Least common multiple of 3 and 8 is 24. Convert \frac{4}{3} and \frac{5}{8} to fractions with denominator 24.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{32}{24}-\frac{15}{24}+\frac{3}{24}\right)}$$

Since \frac{32}{24} and \frac{15}{24} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{32-15}{24}+\frac{3}{24}\right)}$$

Subtract 15 from 32 to get 17.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{17}{24}+\frac{3}{24}\right)}$$

Reduce the fraction \frac{3}{24} to lowest terms by extracting and canceling out 3.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{17}{24}+\frac{1}{8}\right)}$$

Least common multiple of 24 and 8 is 24. Convert \frac{17}{24} and \frac{1}{8} to fractions with denominator 24.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\left(\frac{17}{24}+\frac{3}{24}\right)}$$

Since \frac{17}{24} and \frac{3}{24} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\times \frac{17+3}{24}}$$

Add 17 and 3 to get 20.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\times \frac{20}{24}}$$

Reduce the fraction \frac{20}{24} to lowest terms by extracting and canceling out 4.

$$\frac{25}{12}x=\sqrt{\frac{125}{6}\times \frac{5}{6}}$$

Multiply \frac{125}{6} times \frac{5}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{12}x=\sqrt{\frac{125\times 5}{6\times 6}}$$

Do the multiplications in the fraction \frac{125\times 5}{6\times 6}.

$$\frac{25}{12}x=\sqrt{\frac{625}{36}}$$

Rewrite the square root of the division \frac{625}{36} as the division of square roots \frac{\sqrt{625}}{\sqrt{36}}. Take the square root of both numerator and denominator.

$$\frac{25}{12}x=\frac{25}{6}$$

Multiply both sides by \frac{12}{25}, the reciprocal of \frac{25}{12}.

$$x=\frac{25}{6}\times \frac{12}{25}$$

Multiply \frac{25}{6} times \frac{12}{25} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{25\times 12}{6\times 25}$$

Cancel out 25 in both numerator and denominator.

$$x=\frac{12}{6}$$

Divide 12 by 6 to get 2.

$$x=2$$