Solve (1/1-2/3)*(3/4x-7/2)=frac{5}{6}-x-frac{1}{1}(frac{1}{3}x-frac{5}{1})

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

Solve for x

$$x = \frac{84}{19} = 4\frac{8}{19} \approx 4.421052632$$

Steps for Solving Linear Equation

Anything divided by one gives itself.

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

Convert 1 to fraction \frac{3}{3}.

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

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

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

Subtract 2 from 3 to get 1.

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

Use the distributive property to multiply \frac{1}{3} by \frac{3}{4}x-\frac{7}{2}.

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

Multiply \frac{1}{3} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 3 in both numerator and denominator.

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

Multiply \frac{1}{3} times -\frac{7}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\left(-7\right)}{3\times 2}.

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

Fraction \frac{-7}{6} can be rewritten as -\frac{7}{6} by extracting the negative sign.

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

Anything divided by one gives itself.

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

Anything divided by one gives itself.

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

To find the opposite of \frac{1}{3}x-5, find the opposite of each term.

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

The opposite of -5 is 5.

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

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

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

Convert 5 to fraction \frac{30}{6}.

$$\frac{1}{4}x-\frac{7}{6}=\frac{5}{6}-\frac{4}{3}x+\frac{30}{6}$$

Since \frac{5}{6} and \frac{30}{6} have the same denominator, add them by adding their numerators.

$$\frac{1}{4}x-\frac{7}{6}=\frac{5+30}{6}-\frac{4}{3}x$$

Add 5 and 30 to get 35.

$$\frac{1}{4}x-\frac{7}{6}=\frac{35}{6}-\frac{4}{3}x$$

Add \frac{4}{3}x to both sides.

$$\frac{1}{4}x-\frac{7}{6}+\frac{4}{3}x=\frac{35}{6}$$

Combine \frac{1}{4}x and \frac{4}{3}x to get \frac{19}{12}x.

$$\frac{19}{12}x-\frac{7}{6}=\frac{35}{6}$$

Add \frac{7}{6} to both sides.

$$\frac{19}{12}x=\frac{35}{6}+\frac{7}{6}$$

Since \frac{35}{6} and \frac{7}{6} have the same denominator, add them by adding their numerators.

$$\frac{19}{12}x=\frac{35+7}{6}$$

Add 35 and 7 to get 42.

$$\frac{19}{12}x=\frac{42}{6}$$

Divide 42 by 6 to get 7.

$$\frac{19}{12}x=7$$

Multiply both sides by \frac{12}{19}, the reciprocal of \frac{19}{12}.

$$x=7\times \frac{12}{19}$$

Express 7\times \frac{12}{19} as a single fraction.

$$x=\frac{7\times 12}{19}$$

Multiply 7 and 12 to get 84.

$$x=\frac{84}{19}$$