Solve (1/2x+sqrt{1/9}-2/3*5/4=frac{15}{3}:frac{2}{5}-frac{1}{6}x+frac{1}{2})

$$( \frac { 1 } { 2 } x + \sqrt { \frac { 1 } { 9 } } - \frac { 2 } { 3 } \cdot \frac { 5 } { 4 } = \frac { 15 } { 3 } : \frac { 2 } { 5 } - \frac { 1 } { 6 } x + \frac { 1 } { 2 } )$$

Solve for x

$$x = \frac{81}{4} = 20\frac{1}{4} = 20.25$$

Steps for Solving Linear Equation

Rewrite the square root of the division \frac{1}{9} as the division of square roots \frac{\sqrt{1}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\frac{1}{2}x+\frac{1}{3}-\frac{2}{3}\times \frac{5}{4}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Multiply \frac{2}{3} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{2}x+\frac{1}{3}-\frac{2\times 5}{3\times 4}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Do the multiplications in the fraction \frac{2\times 5}{3\times 4}.

$$\frac{1}{2}x+\frac{1}{3}-\frac{10}{12}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Reduce the fraction \frac{10}{12} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}x+\frac{1}{3}-\frac{5}{6}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Least common multiple of 3 and 6 is 6. Convert \frac{1}{3} and \frac{5}{6} to fractions with denominator 6.

$$\frac{1}{2}x+\frac{2}{6}-\frac{5}{6}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Since \frac{2}{6} and \frac{5}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}x+\frac{2-5}{6}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Subtract 5 from 2 to get -3.

$$\frac{1}{2}x+\frac{-3}{6}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}x-\frac{1}{2}=\frac{\frac{15}{3}}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

Divide 15 by 3 to get 5.

$$\frac{1}{2}x-\frac{1}{2}=\frac{5}{\frac{2}{5}}-\frac{1}{6}x+\frac{1}{2}$$

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

$$\frac{1}{2}x-\frac{1}{2}=5\times \frac{5}{2}-\frac{1}{6}x+\frac{1}{2}$$

Express 5\times \frac{5}{2} as a single fraction.

$$\frac{1}{2}x-\frac{1}{2}=\frac{5\times 5}{2}-\frac{1}{6}x+\frac{1}{2}$$

Multiply 5 and 5 to get 25.

$$\frac{1}{2}x-\frac{1}{2}=\frac{25}{2}-\frac{1}{6}x+\frac{1}{2}$$

Since \frac{25}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}x-\frac{1}{2}=\frac{25+1}{2}-\frac{1}{6}x$$

Add 25 and 1 to get 26.

$$\frac{1}{2}x-\frac{1}{2}=\frac{26}{2}-\frac{1}{6}x$$

Divide 26 by 2 to get 13.

$$\frac{1}{2}x-\frac{1}{2}=13-\frac{1}{6}x$$

Add \frac{1}{6}x to both sides.

$$\frac{1}{2}x-\frac{1}{2}+\frac{1}{6}x=13$$

Combine \frac{1}{2}x and \frac{1}{6}x to get \frac{2}{3}x.

$$\frac{2}{3}x-\frac{1}{2}=13$$

Add \frac{1}{2} to both sides.

$$\frac{2}{3}x=13+\frac{1}{2}$$

Convert 13 to fraction \frac{26}{2}.

$$\frac{2}{3}x=\frac{26}{2}+\frac{1}{2}$$

Since \frac{26}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{2}{3}x=\frac{26+1}{2}$$

Add 26 and 1 to get 27.

$$\frac{2}{3}x=\frac{27}{2}$$

Multiply both sides by \frac{3}{2}, the reciprocal of \frac{2}{3}.

$$x=\frac{27}{2}\times \frac{3}{2}$$

Multiply \frac{27}{2} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{27\times 3}{2\times 2}$$

Do the multiplications in the fraction \frac{27\times 3}{2\times 2}.

$$x=\frac{81}{4}$$