Solve 2/3left(6x/10-5/6right)-3/5left(frac{25}{9}+frac{5x}{6}right)=frac{3x}{5}-frac{1}{2}left(frac{8x}{5}+frac{4}{3}right)

$$\frac{ 2 }{ 3 } \left( \frac{ 6x }{ 10 } - \frac{ 5 }{ 6 } \right) - \frac{ 3 }{ 5 } \left( \frac{ 25 }{ 9 } + \frac{ 5x }{ 6 } \right) = \frac{ 3x }{ 5 } - \frac{ 1 }{ 2 } \left( \frac{ 8x }{ 5 } + \frac{ 4 }{ 3 } \right)$$

Solve for x

$$x = \frac{140}{9} = 15\frac{5}{9} \approx 15.555555556$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 90, the least common multiple of 3,10,6,5,9,2.

$$60\left(\frac{6x}{10}-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Divide 6x by 10 to get \frac{3}{5}x.

$$60\left(\frac{3}{5}x-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Use the distributive property to multiply 60 by \frac{3}{5}x-\frac{5}{6}.

$$60\times \frac{3}{5}x+60\left(-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Express 60\times \frac{3}{5} as a single fraction.

$$\frac{60\times 3}{5}x+60\left(-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Multiply 60 and 3 to get 180.

$$\frac{180}{5}x+60\left(-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Divide 180 by 5 to get 36.

$$36x+60\left(-\frac{5}{6}\right)-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Express 60\left(-\frac{5}{6}\right) as a single fraction.

$$36x+\frac{60\left(-5\right)}{6}-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Multiply 60 and -5 to get -300.

$$36x+\frac{-300}{6}-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Divide -300 by 6 to get -50.

$$36x-50-54\left(\frac{25}{9}+\frac{5x}{6}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 6 is 18. Multiply \frac{25}{9} times \frac{2}{2}. Multiply \frac{5x}{6} times \frac{3}{3}.

$$36x-50-54\left(\frac{25\times 2}{18}+\frac{3\times 5x}{18}\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Since \frac{25\times 2}{18} and \frac{3\times 5x}{18} have the same denominator, add them by adding their numerators.

$$36x-50-54\times \frac{25\times 2+3\times 5x}{18}=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Do the multiplications in 25\times 2+3\times 5x.

$$36x-50-54\times \frac{50+15x}{18}=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Cancel out 18, the greatest common factor in 54 and 18.

$$36x-50-3\left(50+15x\right)=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Use the distributive property to multiply -3 by 50+15x.

$$36x-50-150-45x=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Subtract 150 from -50 to get -200.

$$36x-200-45x=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Combine 36x and -45x to get -9x.

$$-9x-200=18\times 3x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

Multiply 18 and 3 to get 54.

$$-9x-200=54x-45\left(\frac{8x}{5}+\frac{4}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{8x}{5} times \frac{3}{3}. Multiply \frac{4}{3} times \frac{5}{5}.

$$-9x-200=54x-45\left(\frac{3\times 8x}{15}+\frac{4\times 5}{15}\right)$$

Since \frac{3\times 8x}{15} and \frac{4\times 5}{15} have the same denominator, add them by adding their numerators.

$$-9x-200=54x-45\times \frac{3\times 8x+4\times 5}{15}$$

Do the multiplications in 3\times 8x+4\times 5.

$$-9x-200=54x-45\times \frac{24x+20}{15}$$

Cancel out 15, the greatest common factor in 45 and 15.

$$-9x-200=54x-3\left(24x+20\right)$$

Use the distributive property to multiply -3 by 24x+20.

$$-9x-200=54x-72x-60$$

Combine 54x and -72x to get -18x.

$$-9x-200=-18x-60$$

Add 18x to both sides.

$$-9x-200+18x=-60$$

Combine -9x and 18x to get 9x.

$$9x-200=-60$$

Add 200 to both sides.

$$9x=-60+200$$

Add -60 and 200 to get 140.

$$9x=140$$

Divide both sides by 9.

$$x=\frac{140}{9}$$