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

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

Solve for x

$$x=\frac{21}{136}\approx 0.154411765$$

Steps for Solving Linear Equation

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

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

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

$$180\left(\frac{20}{15}+\frac{12}{15}\right)-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Since \frac{20}{15} and \frac{12}{15} have the same denominator, add them by adding their numerators.

$$180\times \frac{20+12}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Add 20 and 12 to get 32.

$$180\times \frac{32}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Express 180\times \frac{32}{15} as a single fraction.

$$\frac{180\times 32}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Multiply 180 and 32 to get 5760.

$$\frac{5760}{15}-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Divide 5760 by 15 to get 384.

$$384-270\left(\frac{8x}{15}+\frac{4}{9}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

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}.

$$384-270\left(\frac{3\times 8x}{45}+\frac{4\times 5}{45}\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

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

$$384-270\times \frac{3\times 8x+4\times 5}{45}=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

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

$$384-270\times \frac{24x+20}{45}=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

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

$$384-6\left(24x+20\right)=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

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

$$384-144x-120=315-240\left(\frac{5}{5}-\frac{9x}{2}\right)$$

Subtract 120 from 384 to get 264.

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

Divide 5 by 5 to get 1.

$$264-144x=315-240\left(1-\frac{9x}{2}\right)$$

Add 240\left(1-\frac{9x}{2}\right) to both sides.

$$264-144x+240\left(1-\frac{9x}{2}\right)=315$$

Use the distributive property to multiply 240 by 1-\frac{9x}{2}.

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

Cancel out 2, the greatest common factor in 240 and 2.

$$264-144x+240-120\times 9x=315$$

Multiply -120 and 9 to get -1080.

$$264-144x+240-1080x=315$$

Add 264 and 240 to get 504.

$$504-144x-1080x=315$$

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

$$504-1224x=315$$

Subtract 504 from both sides.

$$-1224x=315-504$$

Subtract 504 from 315 to get -189.

$$-1224x=-189$$

Divide both sides by -1224.

$$x=\frac{-189}{-1224}$$

Reduce the fraction \frac{-189}{-1224} to lowest terms by extracting and canceling out -9.

$$x=\frac{21}{136}$$