Solve 1/5x-[3/2x-x-1/10-1/100(10x-40)]=frac{x-1}{5}-frac{3-x}{2}

$$\frac { 1 } { 5 } x - [ \frac { 3 } { 2 } x - \frac { x - 1 } { 10 } - \frac { 1 } { 100 } ( 10 x - 40 ) ] = \frac { x - 1 } { 5 } - \frac { 3 - x } { 2 }$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 100, the least common multiple of 5,2,10,100.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{100}\left(10x-40\right)\right)=20\left(x-1\right)-50\left(3-x\right)$$

Use the distributive property to multiply -\frac{1}{100} by 10x-40.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{100}\times 10x-\frac{1}{100}\left(-40\right)\right)=20\left(x-1\right)-50\left(3-x\right)$$

Express -\frac{1}{100}\times 10 as a single fraction.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}+\frac{-10}{100}x-\frac{1}{100}\left(-40\right)\right)=20\left(x-1\right)-50\left(3-x\right)$$

Reduce the fraction \frac{-10}{100} to lowest terms by extracting and canceling out 10.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{10}x-\frac{1}{100}\left(-40\right)\right)=20\left(x-1\right)-50\left(3-x\right)$$

Express -\frac{1}{100}\left(-40\right) as a single fraction.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{10}x+\frac{-\left(-40\right)}{100}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Multiply -1 and -40 to get 40.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{10}x+\frac{40}{100}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Reduce the fraction \frac{40}{100} to lowest terms by extracting and canceling out 20.

$$20x-100\left(\frac{3}{2}x-\frac{x-1}{10}-\frac{1}{10}x+\frac{2}{5}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Combine \frac{3}{2}x and -\frac{1}{10}x to get \frac{7}{5}x.

$$20x-100\left(\frac{7}{5}x-\frac{x-1}{10}+\frac{2}{5}\right)=20\left(x-1\right)-50\left(3-x\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10 and 5 is 10. Multiply \frac{2}{5} times \frac{2}{2}.

$$20x-100\left(\frac{7}{5}x-\frac{x-1}{10}+\frac{2\times 2}{10}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Since -\frac{x-1}{10} and \frac{2\times 2}{10} have the same denominator, add them by adding their numerators.

$$20x-100\left(\frac{7}{5}x+\frac{-\left(x-1\right)+2\times 2}{10}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Do the multiplications in -\left(x-1\right)+2\times 2.

$$20x-100\left(\frac{7}{5}x+\frac{-x+1+4}{10}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Combine like terms in -x+1+4.

$$20x-100\left(\frac{7}{5}x+\frac{-x+5}{10}\right)=20\left(x-1\right)-50\left(3-x\right)$$

Use the distributive property to multiply 20 by x-1.

$$20x-100\left(\frac{7}{5}x+\frac{-x+5}{10}\right)=20x-20-50\left(3-x\right)$$

Use the distributive property to multiply -50 by 3-x.

$$20x-100\left(\frac{7}{5}x+\frac{-x+5}{10}\right)=20x-20-150+50x$$

Subtract 150 from -20 to get -170.

$$20x-100\left(\frac{7}{5}x+\frac{-x+5}{10}\right)=20x-170+50x$$

Combine 20x and 50x to get 70x.

$$20x-100\left(\frac{7}{5}x+\frac{-x+5}{10}\right)=70x-170$$

Divide each term of -x+5 by 10 to get -\frac{1}{10}x+\frac{1}{2}.

$$20x-100\left(\frac{7}{5}x-\frac{1}{10}x+\frac{1}{2}\right)=70x-170$$

Combine \frac{7}{5}x and -\frac{1}{10}x to get \frac{13}{10}x.

$$20x-100\left(\frac{13}{10}x+\frac{1}{2}\right)=70x-170$$

Use the distributive property to multiply -100 by \frac{13}{10}x+\frac{1}{2}.

$$20x-100\times \frac{13}{10}x-100\times \frac{1}{2}=70x-170$$

Express -100\times \frac{13}{10} as a single fraction.

$$20x+\frac{-100\times 13}{10}x-100\times \frac{1}{2}=70x-170$$

Multiply -100 and 13 to get -1300.

$$20x+\frac{-1300}{10}x-100\times \frac{1}{2}=70x-170$$

Divide -1300 by 10 to get -130.

$$20x-130x-100\times \frac{1}{2}=70x-170$$

Multiply -100 and \frac{1}{2} to get \frac{-100}{2}.

$$20x-130x+\frac{-100}{2}=70x-170$$

Divide -100 by 2 to get -50.

$$20x-130x-50=70x-170$$

Combine 20x and -130x to get -110x.

$$-110x-50=70x-170$$

Subtract 70x from both sides.

$$-110x-50-70x=-170$$

Combine -110x and -70x to get -180x.

$$-180x-50=-170$$

Add 50 to both sides.

$$-180x=-170+50$$

Add -170 and 50 to get -120.

$$-180x=-120$$

Divide both sides by -180.

$$x=\frac{-120}{-180}$$

Reduce the fraction \frac{-120}{-180} to lowest terms by extracting and canceling out -60.

$$x=\frac{2}{3}$$