Solve 1/2(3x-1/4)-1/3left(4x-1/5right)=frac{1}{4}left(5x-frac{1}{16}right)+frac{1}{12}

$$\frac{ 1 }{ 2 } (3x- \frac{ 1 }{ 4 } )- \frac{ 1 }{ 3 } \left( 4x- \frac{ 1 }{ 5 } \right) = \frac{ 1 }{ 4 } \left( 5x- \frac{ 1 }{ 16 } \right) + \frac{ 1 }{ 12 }$$

Solve for x

$$x=-\frac{121}{1040}\approx -0.116346154$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}\times 3x+\frac{1}{2}\left(-\frac{1}{4}\right)-\frac{1}{3}\left(4x-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

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

$$\frac{3}{2}x+\frac{1}{2}\left(-\frac{1}{4}\right)-\frac{1}{3}\left(4x-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Multiply \frac{1}{2} times -\frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{2}x+\frac{1\left(-1\right)}{2\times 4}-\frac{1}{3}\left(4x-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 4}.

$$\frac{3}{2}x+\frac{-1}{8}-\frac{1}{3}\left(4x-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Fraction \frac{-1}{8} can be rewritten as -\frac{1}{8} by extracting the negative sign.

$$\frac{3}{2}x-\frac{1}{8}-\frac{1}{3}\left(4x-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Use the distributive property to multiply -\frac{1}{3} by 4x-\frac{1}{5}.

$$\frac{3}{2}x-\frac{1}{8}-\frac{1}{3}\times 4x-\frac{1}{3}\left(-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Express -\frac{1}{3}\times 4 as a single fraction.

$$\frac{3}{2}x-\frac{1}{8}+\frac{-4}{3}x-\frac{1}{3}\left(-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Fraction \frac{-4}{3} can be rewritten as -\frac{4}{3} by extracting the negative sign.

$$\frac{3}{2}x-\frac{1}{8}-\frac{4}{3}x-\frac{1}{3}\left(-\frac{1}{5}\right)=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Multiply -\frac{1}{3} times -\frac{1}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{2}x-\frac{1}{8}-\frac{4}{3}x+\frac{-\left(-1\right)}{3\times 5}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{3\times 5}.

$$\frac{3}{2}x-\frac{1}{8}-\frac{4}{3}x+\frac{1}{15}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Combine \frac{3}{2}x and -\frac{4}{3}x to get \frac{1}{6}x.

$$\frac{1}{6}x-\frac{1}{8}+\frac{1}{15}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Least common multiple of 8 and 15 is 120. Convert -\frac{1}{8} and \frac{1}{15} to fractions with denominator 120.

$$\frac{1}{6}x-\frac{15}{120}+\frac{8}{120}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Since -\frac{15}{120} and \frac{8}{120} have the same denominator, add them by adding their numerators.

$$\frac{1}{6}x+\frac{-15+8}{120}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Add -15 and 8 to get -7.

$$\frac{1}{6}x-\frac{7}{120}=\frac{1}{4}\left(5x-\frac{1}{16}\right)+\frac{1}{12}$$

Use the distributive property to multiply \frac{1}{4} by 5x-\frac{1}{16}.

$$\frac{1}{6}x-\frac{7}{120}=\frac{1}{4}\times 5x+\frac{1}{4}\left(-\frac{1}{16}\right)+\frac{1}{12}$$

Multiply \frac{1}{4} and 5 to get \frac{5}{4}.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x+\frac{1}{4}\left(-\frac{1}{16}\right)+\frac{1}{12}$$

Multiply \frac{1}{4} times -\frac{1}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x+\frac{1\left(-1\right)}{4\times 16}+\frac{1}{12}$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{4\times 16}.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x+\frac{-1}{64}+\frac{1}{12}$$

Fraction \frac{-1}{64} can be rewritten as -\frac{1}{64} by extracting the negative sign.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x-\frac{1}{64}+\frac{1}{12}$$

Least common multiple of 64 and 12 is 192. Convert -\frac{1}{64} and \frac{1}{12} to fractions with denominator 192.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x-\frac{3}{192}+\frac{16}{192}$$

Since -\frac{3}{192} and \frac{16}{192} have the same denominator, add them by adding their numerators.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x+\frac{-3+16}{192}$$

Add -3 and 16 to get 13.

$$\frac{1}{6}x-\frac{7}{120}=\frac{5}{4}x+\frac{13}{192}$$

Subtract \frac{5}{4}x from both sides.

$$\frac{1}{6}x-\frac{7}{120}-\frac{5}{4}x=\frac{13}{192}$$

Combine \frac{1}{6}x and -\frac{5}{4}x to get -\frac{13}{12}x.

$$-\frac{13}{12}x-\frac{7}{120}=\frac{13}{192}$$

Add \frac{7}{120} to both sides.

$$-\frac{13}{12}x=\frac{13}{192}+\frac{7}{120}$$

Least common multiple of 192 and 120 is 960. Convert \frac{13}{192} and \frac{7}{120} to fractions with denominator 960.

$$-\frac{13}{12}x=\frac{65}{960}+\frac{56}{960}$$

Since \frac{65}{960} and \frac{56}{960} have the same denominator, add them by adding their numerators.

$$-\frac{13}{12}x=\frac{65+56}{960}$$

Add 65 and 56 to get 121.

$$-\frac{13}{12}x=\frac{121}{960}$$

Multiply both sides by -\frac{12}{13}, the reciprocal of -\frac{13}{12}.

$$x=\frac{121}{960}\left(-\frac{12}{13}\right)$$

Multiply \frac{121}{960} times -\frac{12}{13} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{121\left(-12\right)}{960\times 13}$$

Do the multiplications in the fraction \frac{121\left(-12\right)}{960\times 13}.

$$x=\frac{-1452}{12480}$$

Reduce the fraction \frac{-1452}{12480} to lowest terms by extracting and canceling out 12.

$$x=-\frac{121}{1040}$$