Solve 12left(3x/4-7/2right)-6left(5/9+3x/4right)=frac{3x}{2}+18left(frac{5x}{6}-frac{7}{12}right)

$$12 \left( \frac{ 3x }{ 4 } - \frac{ 7 }{ 2 } \right) -6 \left( \frac{ 5 }{ 9 } + \frac{ 3x }{ 4 } \right) = \frac{ 3x }{ 2 } +18 \left( \frac{ 5x }{ 6 } - \frac{ 7 }{ 12 } \right)$$

Solve for x

$$x = -\frac{209}{72} = -2\frac{65}{72} \approx -2.902777778$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 36, the least common multiple of 4,2,9,6,12.

$$432\left(\frac{3x}{4}-\frac{7}{2}\right)-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{7}{2} times \frac{2}{2}.

$$432\left(\frac{3x}{4}-\frac{7\times 2}{4}\right)-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Since \frac{3x}{4} and \frac{7\times 2}{4} have the same denominator, subtract them by subtracting their numerators.

$$432\times \frac{3x-7\times 2}{4}-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Do the multiplications in 3x-7\times 2.

$$432\times \frac{3x-14}{4}-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Cancel out 4, the greatest common factor in 432 and 4.

$$108\left(3x-14\right)-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Use the distributive property to multiply 108 by 3x-14.

$$324x-1512-216\left(\frac{5}{9}+\frac{3x}{4}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 4 is 36. Multiply \frac{5}{9} times \frac{4}{4}. Multiply \frac{3x}{4} times \frac{9}{9}.

$$324x-1512-216\left(\frac{5\times 4}{36}+\frac{9\times 3x}{36}\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Since \frac{5\times 4}{36} and \frac{9\times 3x}{36} have the same denominator, add them by adding their numerators.

$$324x-1512-216\times \frac{5\times 4+9\times 3x}{36}=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Do the multiplications in 5\times 4+9\times 3x.

$$324x-1512-216\times \frac{20+27x}{36}=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Cancel out 36, the greatest common factor in 216 and 36.

$$324x-1512-6\left(20+27x\right)=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Use the distributive property to multiply -6 by 20+27x.

$$324x-1512-120-162x=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Subtract 120 from -1512 to get -1632.

$$324x-1632-162x=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Combine 324x and -162x to get 162x.

$$162x-1632=18\times 3x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

Multiply 18 and 3 to get 54.

$$162x-1632=54x+648\left(\frac{5x}{6}-\frac{7}{12}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 12 is 12. Multiply \frac{5x}{6} times \frac{2}{2}.

$$162x-1632=54x+648\left(\frac{2\times 5x}{12}-\frac{7}{12}\right)$$

Since \frac{2\times 5x}{12} and \frac{7}{12} have the same denominator, subtract them by subtracting their numerators.

$$162x-1632=54x+648\times \frac{2\times 5x-7}{12}$$

Do the multiplications in 2\times 5x-7.

$$162x-1632=54x+648\times \frac{10x-7}{12}$$

Cancel out 12, the greatest common factor in 648 and 12.

$$162x-1632=54x+54\left(10x-7\right)$$

Use the distributive property to multiply 54 by 10x-7.

$$162x-1632=54x+540x-378$$

Combine 54x and 540x to get 594x.

$$162x-1632=594x-378$$

Subtract 594x from both sides.

$$162x-1632-594x=-378$$

Combine 162x and -594x to get -432x.

$$-432x-1632=-378$$

Add 1632 to both sides.

$$-432x=-378+1632$$

Add -378 and 1632 to get 1254.

$$-432x=1254$$

Divide both sides by -432.

$$x=\frac{1254}{-432}$$

Reduce the fraction \frac{1254}{-432} to lowest terms by extracting and canceling out 6.

$$x=-\frac{209}{72}$$