Solve 4/7+1/2+1/10/1+frac{2{5}}:(frac{3}{4}+frac{2}{3}-frac{1}{12})=[frac{frac{2}{3}-frac{1}{3}+frac{3}{4}}{frac{13}{4}}]:x

$$\frac { 4 } { 7 } + \frac { \frac { 1 } { 2 } + \frac { 1 } { 10 } } { 1 + \frac { 2 } { 5 } } : ( \frac { 3 } { 4 } + \frac { 2 } { 3 } - \frac { 1 } { 12 } ) = [ \frac { \frac { 2 } { 3 } - \frac { 1 } { 3 } + \frac { 3 } { 4 } } { \frac { 13 } { 4 } } ] : x$$

Solve for x

$$x=\frac{28}{75}\approx 0.373333333$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of 7,x.

$$7x\times \frac{4}{7}+\frac{21}{4}x\times \frac{\frac{1}{2}+\frac{1}{10}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Cancel out 7 and 7.

$$4x+\frac{21}{4}x\times \frac{\frac{1}{2}+\frac{1}{10}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

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

$$4x+\frac{21}{4}x\times \frac{\frac{5}{10}+\frac{1}{10}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Since \frac{5}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$4x+\frac{21}{4}x\times \frac{\frac{5+1}{10}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Add 5 and 1 to get 6.

$$4x+\frac{21}{4}x\times \frac{\frac{6}{10}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.

$$4x+\frac{21}{4}x\times \frac{\frac{3}{5}}{1+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Convert 1 to fraction \frac{5}{5}.

$$4x+\frac{21}{4}x\times \frac{\frac{3}{5}}{\frac{5}{5}+\frac{2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

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

$$4x+\frac{21}{4}x\times \frac{\frac{3}{5}}{\frac{5+2}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Add 5 and 2 to get 7.

$$4x+\frac{21}{4}x\times \frac{\frac{3}{5}}{\frac{7}{5}}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Divide \frac{3}{5} by \frac{7}{5} by multiplying \frac{3}{5} by the reciprocal of \frac{7}{5}.

$$4x+\frac{21}{4}x\times \frac{3}{5}\times \frac{5}{7}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Multiply \frac{3}{5} times \frac{5}{7} by multiplying numerator times numerator and denominator times denominator.

$$4x+\frac{21}{4}x\times \frac{3\times 5}{5\times 7}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Cancel out 5 in both numerator and denominator.

$$4x+\frac{21}{4}x\times \frac{3}{7}=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Multiply \frac{21}{4} times \frac{3}{7} by multiplying numerator times numerator and denominator times denominator.

$$4x+\frac{21\times 3}{4\times 7}x=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Do the multiplications in the fraction \frac{21\times 3}{4\times 7}.

$$4x+\frac{63}{28}x=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Reduce the fraction \frac{63}{28} to lowest terms by extracting and canceling out 7.

$$4x+\frac{9}{4}x=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Combine 4x and \frac{9}{4}x to get \frac{25}{4}x.

$$\frac{25}{4}x=7\times \frac{\frac{2}{3}-\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Since \frac{2}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25}{4}x=7\times \frac{\frac{2-1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

Subtract 1 from 2 to get 1.

$$\frac{25}{4}x=7\times \frac{\frac{1}{3}+\frac{3}{4}}{\frac{13}{4}}$$

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

$$\frac{25}{4}x=7\times \frac{\frac{4}{12}+\frac{9}{12}}{\frac{13}{4}}$$

Since \frac{4}{12} and \frac{9}{12} have the same denominator, add them by adding their numerators.

$$\frac{25}{4}x=7\times \frac{\frac{4+9}{12}}{\frac{13}{4}}$$

Add 4 and 9 to get 13.

$$\frac{25}{4}x=7\times \frac{\frac{13}{12}}{\frac{13}{4}}$$

Divide \frac{13}{12} by \frac{13}{4} by multiplying \frac{13}{12} by the reciprocal of \frac{13}{4}.

$$\frac{25}{4}x=7\times \frac{13}{12}\times \frac{4}{13}$$

Multiply \frac{13}{12} times \frac{4}{13} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25}{4}x=7\times \frac{13\times 4}{12\times 13}$$

Cancel out 13 in both numerator and denominator.

$$\frac{25}{4}x=7\times \frac{4}{12}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

$$\frac{25}{4}x=7\times \frac{1}{3}$$

Multiply 7 and \frac{1}{3} to get \frac{7}{3}.

$$\frac{25}{4}x=\frac{7}{3}$$

Multiply both sides by \frac{4}{25}, the reciprocal of \frac{25}{4}.

$$x=\frac{7}{3}\times \frac{4}{25}$$

Multiply \frac{7}{3} times \frac{4}{25} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{7\times 4}{3\times 25}$$

Do the multiplications in the fraction \frac{7\times 4}{3\times 25}.

$$x=\frac{28}{75}$$