Solve 3/4x+1/2-2(1/3-x)=1/3-x+3(x-2+frac{1}{12})

$$\frac { 3 } { 4 } x + \frac { 1 } { 2 } - 2 ( \frac { 1 } { 3 } - x ) = \frac { 1 } { 3 } - x + 3 ( x - 2 + \frac { 1 } { 12 } )$$

Solve for x

$$x=-7$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2 by \frac{1}{3}-x.

$$\frac{3}{4}x+\frac{1}{2}-2\times \frac{1}{3}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Multiply -2 and \frac{1}{3} to get \frac{-2}{3}.

$$\frac{3}{4}x+\frac{1}{2}+\frac{-2}{3}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{3}{4}x+\frac{1}{2}-\frac{2}{3}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{2}{3} to fractions with denominator 6.

$$\frac{3}{4}x+\frac{3}{6}-\frac{4}{6}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Since \frac{3}{6} and \frac{4}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{4}x+\frac{3-4}{6}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Subtract 4 from 3 to get -1.

$$\frac{3}{4}x-\frac{1}{6}+2x=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Combine \frac{3}{4}x and 2x to get \frac{11}{4}x.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3\left(x-2+\frac{1}{12}\right)$$

Convert -2 to fraction -\frac{24}{12}.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3\left(x-\frac{24}{12}+\frac{1}{12}\right)$$

Since -\frac{24}{12} and \frac{1}{12} have the same denominator, add them by adding their numerators.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3\left(x+\frac{-24+1}{12}\right)$$

Add -24 and 1 to get -23.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3\left(x-\frac{23}{12}\right)$$

Use the distributive property to multiply 3 by x-\frac{23}{12}.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3x+3\left(-\frac{23}{12}\right)$$

Express 3\left(-\frac{23}{12}\right) as a single fraction.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3x+\frac{3\left(-23\right)}{12}$$

Multiply 3 and -23 to get -69.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3x+\frac{-69}{12}$$

Reduce the fraction \frac{-69}{12} to lowest terms by extracting and canceling out 3.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}-x+3x-\frac{23}{4}$$

Combine -x and 3x to get 2x.

$$\frac{11}{4}x-\frac{1}{6}=\frac{1}{3}+2x-\frac{23}{4}$$

Least common multiple of 3 and 4 is 12. Convert \frac{1}{3} and \frac{23}{4} to fractions with denominator 12.

$$\frac{11}{4}x-\frac{1}{6}=\frac{4}{12}+2x-\frac{69}{12}$$

Since \frac{4}{12} and \frac{69}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{11}{4}x-\frac{1}{6}=\frac{4-69}{12}+2x$$

Subtract 69 from 4 to get -65.

$$\frac{11}{4}x-\frac{1}{6}=-\frac{65}{12}+2x$$

Subtract 2x from both sides.

$$\frac{11}{4}x-\frac{1}{6}-2x=-\frac{65}{12}$$

Combine \frac{11}{4}x and -2x to get \frac{3}{4}x.

$$\frac{3}{4}x-\frac{1}{6}=-\frac{65}{12}$$

Add \frac{1}{6} to both sides.

$$\frac{3}{4}x=-\frac{65}{12}+\frac{1}{6}$$

Least common multiple of 12 and 6 is 12. Convert -\frac{65}{12} and \frac{1}{6} to fractions with denominator 12.

$$\frac{3}{4}x=-\frac{65}{12}+\frac{2}{12}$$

Since -\frac{65}{12} and \frac{2}{12} have the same denominator, add them by adding their numerators.

$$\frac{3}{4}x=\frac{-65+2}{12}$$

Add -65 and 2 to get -63.

$$\frac{3}{4}x=\frac{-63}{12}$$

Reduce the fraction \frac{-63}{12} to lowest terms by extracting and canceling out 3.

$$\frac{3}{4}x=-\frac{21}{4}$$

Multiply both sides by \frac{4}{3}, the reciprocal of \frac{3}{4}.

$$x=-\frac{21}{4}\times \frac{4}{3}$$

Multiply -\frac{21}{4} times \frac{4}{3} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-21\times 4}{4\times 3}$$

Cancel out 4 in both numerator and denominator.

$$x=\frac{-21}{3}$$

Divide -21 by 3 to get -7.

$$x=-7$$