Solve 2/3+4/5-3/2(8x/15+frac{4}{9})=frac{7}{4}-frac{4}{3}(frac{6}{5}-frac{9x}{2})

$$\frac { 2 } { 3 } + \frac { 4 } { 5 } - \frac { 3 } { 2 } ( \frac { 8 x } { 15 } + \frac { 4 } { 9 } ) = \frac { 7 } { 4 } - \frac { 4 } { 3 } ( \frac { 6 } { 5 } - \frac { 9 x } { 2 } )$$

Solve for x

$$x=\frac{13}{136}\approx 0.095588235$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 180, the least common multiple of 3,5,2,15,9,4.

$$180\left(\frac{2}{3}+\frac{4}{5}\right)-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Least common multiple of 3 and 5 is 15. Convert \frac{2}{3} and \frac{4}{5} to fractions with denominator 15.

$$180\left(\frac{10}{15}+\frac{12}{15}\right)-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Since \frac{10}{15} and \frac{12}{15} have the same denominator, add them by adding their numerators.

$$180\times \frac{10+12}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Add 10 and 12 to get 22.

$$180\times \frac{22}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Express 180\times \frac{22}{15} as a single fraction.

$$\frac{180\times 22}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Multiply 180 and 22 to get 3960.

$$\frac{3960}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Divide 3960 by 15 to get 264.

$$264-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15 and 9 is 45. Multiply \frac{8x}{15} times \frac{3}{3}. Multiply \frac{4}{9} times \frac{5}{5}.

$$264-270\left(\frac{3\times 8x}{45}+\frac{4\times 5}{45}\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Since \frac{3\times 8x}{45} and \frac{4\times 5}{45} have the same denominator, add them by adding their numerators.

$$264-270\times \frac{3\times 8x+4\times 5}{45}=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

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

$$264-270\times \frac{24x+20}{45}=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Cancel out 45, the greatest common factor in 270 and 45.

$$264-6\left(24x+20\right)=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Use the distributive property to multiply -6 by 24x+20.

$$264-144x-120=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

Subtract 120 from 264 to get 144.

$$144-144x=315-240\left(\frac{6}{5}-\frac{9x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{6}{5} times \frac{2}{2}. Multiply \frac{9x}{2} times \frac{5}{5}.

$$144-144x=315-240\left(\frac{6\times 2}{10}-\frac{5\times 9x}{10}\right)$$

Since \frac{6\times 2}{10} and \frac{5\times 9x}{10} have the same denominator, subtract them by subtracting their numerators.

$$144-144x=315-240\times \frac{6\times 2-5\times 9x}{10}$$

Do the multiplications in 6\times 2-5\times 9x.

$$144-144x=315-240\times \frac{12-45x}{10}$$

Cancel out 10, the greatest common factor in 240 and 10.

$$144-144x=315-24\left(12-45x\right)$$

Use the distributive property to multiply -24 by 12-45x.

$$144-144x=315-288+1080x$$

Subtract 288 from 315 to get 27.

$$144-144x=27+1080x$$

Subtract 1080x from both sides.

$$144-144x-1080x=27$$

Combine -144x and -1080x to get -1224x.

$$144-1224x=27$$

Subtract 144 from both sides.

$$-1224x=27-144$$

Subtract 144 from 27 to get -117.

$$-1224x=-117$$

Divide both sides by -1224.

$$x=\frac{-117}{-1224}$$

Reduce the fraction \frac{-117}{-1224} to lowest terms by extracting and canceling out -9.

$$x=\frac{13}{136}$$