Solve 4/5(15x/2+10/3)-3/4(frac{8x}{9}-frac{5}{6})=frac{5}{6}(30x-frac{2}{15})+frac{2}{15}

$$\frac { 4 } { 5 } ( \frac { 15 x } { 2 } + \frac { 10 } { 3 } ) - \frac { 3 } { 4 } ( \frac { 8 x } { 9 } - \frac { 5 } { 6 } ) = \frac { 5 } { 6 } ( 30 x - \frac { 2 } { 15 } ) + \frac { 2 } { 15 }$$

Solve for x

$$x=\frac{1177}{7080}\approx 0.166242938$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 180, the least common multiple of 5,2,3,4,9,6,15.

$$144\left(\frac{15x}{2}+\frac{10}{3}\right)-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{15x}{2} times \frac{3}{3}. Multiply \frac{10}{3} times \frac{2}{2}.

$$144\left(\frac{3\times 15x}{6}+\frac{10\times 2}{6}\right)-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

Since \frac{3\times 15x}{6} and \frac{10\times 2}{6} have the same denominator, add them by adding their numerators.

$$144\times \frac{3\times 15x+10\times 2}{6}-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

Do the multiplications in 3\times 15x+10\times 2.

$$144\times \frac{45x+20}{6}-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

Cancel out 6, the greatest common factor in 144 and 6.

$$24\left(45x+20\right)-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

Use the distributive property to multiply 24 by 45x+20.

$$1080x+480-135\left(\frac{8x}{9}-\frac{5}{6}\right)=150\left(30x-\frac{2}{15}\right)+24$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 6 is 18. Multiply \frac{8x}{9} times \frac{2}{2}. Multiply \frac{5}{6} times \frac{3}{3}.

$$1080x+480-135\left(\frac{2\times 8x}{18}-\frac{5\times 3}{18}\right)=150\left(30x-\frac{2}{15}\right)+24$$

Since \frac{2\times 8x}{18} and \frac{5\times 3}{18} have the same denominator, subtract them by subtracting their numerators.

$$1080x+480-135\times \frac{2\times 8x-5\times 3}{18}=150\left(30x-\frac{2}{15}\right)+24$$

Do the multiplications in 2\times 8x-5\times 3.

$$1080x+480-135\times \frac{16x-15}{18}=150\left(30x-\frac{2}{15}\right)+24$$

Express 135\times \frac{16x-15}{18} as a single fraction.

$$1080x+480-\frac{135\left(16x-15\right)}{18}=150\left(30x-\frac{2}{15}\right)+24$$

Use the distributive property to multiply 135 by 16x-15.

$$1080x+480-\frac{2160x-2025}{18}=150\left(30x-\frac{2}{15}\right)+24$$

Use the distributive property to multiply 150 by 30x-\frac{2}{15}.

$$1080x+480-\frac{2160x-2025}{18}=4500x+150\left(-\frac{2}{15}\right)+24$$

Express 150\left(-\frac{2}{15}\right) as a single fraction.

$$1080x+480-\frac{2160x-2025}{18}=4500x+\frac{150\left(-2\right)}{15}+24$$

Multiply 150 and -2 to get -300.

$$1080x+480-\frac{2160x-2025}{18}=4500x+\frac{-300}{15}+24$$

Divide -300 by 15 to get -20.

$$1080x+480-\frac{2160x-2025}{18}=4500x-20+24$$

Add -20 and 24 to get 4.

$$1080x+480-\frac{2160x-2025}{18}=4500x+4$$

Divide each term of 2160x-2025 by 18 to get 120x-\frac{225}{2}.

$$1080x+480-\left(120x-\frac{225}{2}\right)=4500x+4$$

To find the opposite of 120x-\frac{225}{2}, find the opposite of each term.

$$1080x+480-120x-\left(-\frac{225}{2}\right)=4500x+4$$

The opposite of -\frac{225}{2} is \frac{225}{2}.

$$1080x+480-120x+\frac{225}{2}=4500x+4$$

Combine 1080x and -120x to get 960x.

$$960x+480+\frac{225}{2}=4500x+4$$

Convert 480 to fraction \frac{960}{2}.

$$960x+\frac{960}{2}+\frac{225}{2}=4500x+4$$

Since \frac{960}{2} and \frac{225}{2} have the same denominator, add them by adding their numerators.

$$960x+\frac{960+225}{2}=4500x+4$$

Add 960 and 225 to get 1185.

$$960x+\frac{1185}{2}=4500x+4$$

Subtract 4500x from both sides.

$$960x+\frac{1185}{2}-4500x=4$$

Combine 960x and -4500x to get -3540x.

$$-3540x+\frac{1185}{2}=4$$

Subtract \frac{1185}{2} from both sides.

$$-3540x=4-\frac{1185}{2}$$

Convert 4 to fraction \frac{8}{2}.

$$-3540x=\frac{8}{2}-\frac{1185}{2}$$

Since \frac{8}{2} and \frac{1185}{2} have the same denominator, subtract them by subtracting their numerators.

$$-3540x=\frac{8-1185}{2}$$

Subtract 1185 from 8 to get -1177.

$$-3540x=-\frac{1177}{2}$$

Divide both sides by -3540.

$$x=\frac{-\frac{1177}{2}}{-3540}$$

Express \frac{-\frac{1177}{2}}{-3540} as a single fraction.

$$x=\frac{-1177}{2\left(-3540\right)}$$

Multiply 2 and -3540 to get -7080.

$$x=\frac{-1177}{-7080}$$

Fraction \frac{-1177}{-7080} can be simplified to \frac{1177}{7080} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{1177}{7080}$$