Solve 1/3(5x+2)-1/6(4x-1)=1/12(-3x-7)+1/6

$$\frac { 1 } { 3 } ( 5 x + 2 ) - \frac { 1 } { 6 } ( 4 x - 1 ) = \frac { 1 } { 12 } ( - 3 x - 7 ) + \frac { 1 } { 6 }$$

Solve for x

$$x=-1$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{3} by 5x+2.

$$\frac{1}{3}\times 5x+\frac{1}{3}\times 2-\frac{1}{6}\left(4x-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Multiply \frac{1}{3} and 5 to get \frac{5}{3}.

$$\frac{5}{3}x+\frac{1}{3}\times 2-\frac{1}{6}\left(4x-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

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

$$\frac{5}{3}x+\frac{2}{3}-\frac{1}{6}\left(4x-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Use the distributive property to multiply -\frac{1}{6} by 4x-1.

$$\frac{5}{3}x+\frac{2}{3}-\frac{1}{6}\times 4x-\frac{1}{6}\left(-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Express -\frac{1}{6}\times 4 as a single fraction.

$$\frac{5}{3}x+\frac{2}{3}+\frac{-4}{6}x-\frac{1}{6}\left(-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$\frac{5}{3}x+\frac{2}{3}-\frac{2}{3}x-\frac{1}{6}\left(-1\right)=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Multiply -\frac{1}{6} and -1 to get \frac{1}{6}.

$$\frac{5}{3}x+\frac{2}{3}-\frac{2}{3}x+\frac{1}{6}=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Combine \frac{5}{3}x and -\frac{2}{3}x to get x.

$$x+\frac{2}{3}+\frac{1}{6}=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

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

$$x+\frac{4}{6}+\frac{1}{6}=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Since \frac{4}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$x+\frac{4+1}{6}=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Add 4 and 1 to get 5.

$$x+\frac{5}{6}=\frac{1}{12}\left(-3x-7\right)+\frac{1}{6}$$

Use the distributive property to multiply \frac{1}{12} by -3x-7.

$$x+\frac{5}{6}=\frac{1}{12}\left(-3\right)x+\frac{1}{12}\left(-7\right)+\frac{1}{6}$$

Multiply \frac{1}{12} and -3 to get \frac{-3}{12}.

$$x+\frac{5}{6}=\frac{-3}{12}x+\frac{1}{12}\left(-7\right)+\frac{1}{6}$$

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

$$x+\frac{5}{6}=-\frac{1}{4}x+\frac{1}{12}\left(-7\right)+\frac{1}{6}$$

Multiply \frac{1}{12} and -7 to get \frac{-7}{12}.

$$x+\frac{5}{6}=-\frac{1}{4}x+\frac{-7}{12}+\frac{1}{6}$$

Fraction \frac{-7}{12} can be rewritten as -\frac{7}{12} by extracting the negative sign.

$$x+\frac{5}{6}=-\frac{1}{4}x-\frac{7}{12}+\frac{1}{6}$$

Least common multiple of 12 and 6 is 12. Convert -\frac{7}{12} and \frac{1}{6} to fractions with denominator 12.

$$x+\frac{5}{6}=-\frac{1}{4}x-\frac{7}{12}+\frac{2}{12}$$

Since -\frac{7}{12} and \frac{2}{12} have the same denominator, add them by adding their numerators.

$$x+\frac{5}{6}=-\frac{1}{4}x+\frac{-7+2}{12}$$

Add -7 and 2 to get -5.

$$x+\frac{5}{6}=-\frac{1}{4}x-\frac{5}{12}$$

Add \frac{1}{4}x to both sides.

$$x+\frac{5}{6}+\frac{1}{4}x=-\frac{5}{12}$$

Combine x and \frac{1}{4}x to get \frac{5}{4}x.

$$\frac{5}{4}x+\frac{5}{6}=-\frac{5}{12}$$

Subtract \frac{5}{6} from both sides.

$$\frac{5}{4}x=-\frac{5}{12}-\frac{5}{6}$$

Least common multiple of 12 and 6 is 12. Convert -\frac{5}{12} and \frac{5}{6} to fractions with denominator 12.

$$\frac{5}{4}x=-\frac{5}{12}-\frac{10}{12}$$

Since -\frac{5}{12} and \frac{10}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{4}x=\frac{-5-10}{12}$$

Subtract 10 from -5 to get -15.

$$\frac{5}{4}x=\frac{-15}{12}$$

Reduce the fraction \frac{-15}{12} to lowest terms by extracting and canceling out 3.

$$\frac{5}{4}x=-\frac{5}{4}$$

Multiply both sides by \frac{4}{5}, the reciprocal of \frac{5}{4}.

$$x=-\frac{5}{4}\times \frac{4}{5}$$

Multiply -\frac{5}{4} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-5\times 4}{4\times 5}$$

Cancel out 4 in both numerator and denominator.

$$x=\frac{-5}{5}$$

Divide -5 by 5 to get -1.

$$x=-1$$