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

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

Solve for x

$$x=-\frac{3}{28}\approx -0.107142857$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{5}{6} by x-3.

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

Express \frac{5}{6}\left(-3\right) as a single fraction.

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

Multiply 5 and -3 to get -15.

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

Reduce the fraction \frac{-15}{6} to lowest terms by extracting and canceling out 3.

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

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

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

Express -\frac{1}{2}\left(-4\right) as a single fraction.

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

Multiply -1 and -4 to get 4.

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

Divide 4 by 2 to get 2.

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

Combine \frac{5}{6}x and -\frac{1}{2}x to get \frac{1}{3}x.

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

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

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

Since -\frac{5}{2} and \frac{4}{2} have the same denominator, add them by adding their numerators.

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

Add -5 and 4 to get -1.

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

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

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

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

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

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

$$\frac{1}{3}x-\frac{1}{2}=\frac{2}{5}x+\frac{-3}{5}-x$$

Fraction \frac{-3}{5} can be rewritten as -\frac{3}{5} by extracting the negative sign.

$$\frac{1}{3}x-\frac{1}{2}=\frac{2}{5}x-\frac{3}{5}-x$$

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

$$\frac{1}{3}x-\frac{1}{2}=-\frac{3}{5}x-\frac{3}{5}$$

Add \frac{3}{5}x to both sides.

$$\frac{1}{3}x-\frac{1}{2}+\frac{3}{5}x=-\frac{3}{5}$$

Combine \frac{1}{3}x and \frac{3}{5}x to get \frac{14}{15}x.

$$\frac{14}{15}x-\frac{1}{2}=-\frac{3}{5}$$

Add \frac{1}{2} to both sides.

$$\frac{14}{15}x=-\frac{3}{5}+\frac{1}{2}$$

Least common multiple of 5 and 2 is 10. Convert -\frac{3}{5} and \frac{1}{2} to fractions with denominator 10.

$$\frac{14}{15}x=-\frac{6}{10}+\frac{5}{10}$$

Since -\frac{6}{10} and \frac{5}{10} have the same denominator, add them by adding their numerators.

$$\frac{14}{15}x=\frac{-6+5}{10}$$

Add -6 and 5 to get -1.

$$\frac{14}{15}x=-\frac{1}{10}$$

Multiply both sides by \frac{15}{14}, the reciprocal of \frac{14}{15}.

$$x=-\frac{1}{10}\times \frac{15}{14}$$

Multiply -\frac{1}{10} times \frac{15}{14} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-15}{10\times 14}$$

Do the multiplications in the fraction \frac{-15}{10\times 14}.

$$x=\frac{-15}{140}$$

Reduce the fraction \frac{-15}{140} to lowest terms by extracting and canceling out 5.

$$x=-\frac{3}{28}$$