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

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

Solve for x

$$x=\frac{23}{124}\approx 0.185483871$$

Steps for Solving Linear Equation

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

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

Multiply \frac{1}{5} and -2 to get \frac{-2}{5}.

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

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

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

To find the opposite of 2x-3, find the opposite of each term.

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

The opposite of -3 is 3.

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

Combine \frac{1}{5}x and -2x to get -\frac{9}{5}x.

$$-\frac{9}{5}x-\frac{2}{5}+3=\frac{2}{3}\left(4x+1\right)-\frac{1}{6}\left(2x-7\right)$$

Convert 3 to fraction \frac{15}{5}.

$$-\frac{9}{5}x-\frac{2}{5}+\frac{15}{5}=\frac{2}{3}\left(4x+1\right)-\frac{1}{6}\left(2x-7\right)$$

Since -\frac{2}{5} and \frac{15}{5} have the same denominator, add them by adding their numerators.

$$-\frac{9}{5}x+\frac{-2+15}{5}=\frac{2}{3}\left(4x+1\right)-\frac{1}{6}\left(2x-7\right)$$

Add -2 and 15 to get 13.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{2}{3}\left(4x+1\right)-\frac{1}{6}\left(2x-7\right)$$

Use the distributive property to multiply \frac{2}{3} by 4x+1.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{2}{3}\times 4x+\frac{2}{3}-\frac{1}{6}\left(2x-7\right)$$

Express \frac{2}{3}\times 4 as a single fraction.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{2\times 4}{3}x+\frac{2}{3}-\frac{1}{6}\left(2x-7\right)$$

Multiply 2 and 4 to get 8.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}-\frac{1}{6}\left(2x-7\right)$$

Use the distributive property to multiply -\frac{1}{6} by 2x-7.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}-\frac{1}{6}\times 2x-\frac{1}{6}\left(-7\right)$$

Express -\frac{1}{6}\times 2 as a single fraction.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}+\frac{-2}{6}x-\frac{1}{6}\left(-7\right)$$

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

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}-\frac{1}{3}x-\frac{1}{6}\left(-7\right)$$

Express -\frac{1}{6}\left(-7\right) as a single fraction.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}-\frac{1}{3}x+\frac{-\left(-7\right)}{6}$$

Multiply -1 and -7 to get 7.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{8}{3}x+\frac{2}{3}-\frac{1}{3}x+\frac{7}{6}$$

Combine \frac{8}{3}x and -\frac{1}{3}x to get \frac{7}{3}x.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{7}{3}x+\frac{2}{3}+\frac{7}{6}$$

Least common multiple of 3 and 6 is 6. Convert \frac{2}{3} and \frac{7}{6} to fractions with denominator 6.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{7}{3}x+\frac{4}{6}+\frac{7}{6}$$

Since \frac{4}{6} and \frac{7}{6} have the same denominator, add them by adding their numerators.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{7}{3}x+\frac{4+7}{6}$$

Add 4 and 7 to get 11.

$$-\frac{9}{5}x+\frac{13}{5}=\frac{7}{3}x+\frac{11}{6}$$

Subtract \frac{7}{3}x from both sides.

$$-\frac{9}{5}x+\frac{13}{5}-\frac{7}{3}x=\frac{11}{6}$$

Combine -\frac{9}{5}x and -\frac{7}{3}x to get -\frac{62}{15}x.

$$-\frac{62}{15}x+\frac{13}{5}=\frac{11}{6}$$

Subtract \frac{13}{5} from both sides.

$$-\frac{62}{15}x=\frac{11}{6}-\frac{13}{5}$$

Least common multiple of 6 and 5 is 30. Convert \frac{11}{6} and \frac{13}{5} to fractions with denominator 30.

$$-\frac{62}{15}x=\frac{55}{30}-\frac{78}{30}$$

Since \frac{55}{30} and \frac{78}{30} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{62}{15}x=\frac{55-78}{30}$$

Subtract 78 from 55 to get -23.

$$-\frac{62}{15}x=-\frac{23}{30}$$

Multiply both sides by -\frac{15}{62}, the reciprocal of -\frac{62}{15}.

$$x=-\frac{23}{30}\left(-\frac{15}{62}\right)$$

Multiply -\frac{23}{30} times -\frac{15}{62} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-23\left(-15\right)}{30\times 62}$$

Do the multiplications in the fraction \frac{-23\left(-15\right)}{30\times 62}.

$$x=\frac{345}{1860}$$

Reduce the fraction \frac{345}{1860} to lowest terms by extracting and canceling out 15.

$$x=\frac{23}{124}$$