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

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

Solve for x

$$x=5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 5,2,3,10.

$$30\left(\frac{3x+2}{5}+\frac{1}{2}x\right)-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Use the distributive property to multiply 30 by \frac{3x+2}{5}+\frac{1}{2}x.

$$30\times \frac{3x+2}{5}+30\times \frac{1}{2}x-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Cancel out 5, the greatest common factor in 30 and 5.

$$6\left(3x+2\right)+30\times \frac{1}{2}x-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Use the distributive property to multiply 6 by 3x+2.

$$18x+12+30\times \frac{1}{2}x-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Multiply 30 and \frac{1}{2} to get \frac{30}{2}.

$$18x+12+\frac{30}{2}x-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Divide 30 by 2 to get 15.

$$18x+12+15x-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

Combine 18x and 15x to get 33x.

$$33x+12-6\left(x+2-\frac{1}{2}\left(x-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

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

$$33x+12-6\left(x+2-\frac{1}{2}x-\frac{1}{2}\left(-\frac{2}{3}\right)\right)=3\left(3x+1\right)+20x$$

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

$$33x+12-6\left(x+2-\frac{1}{2}x+\frac{-\left(-2\right)}{2\times 3}\right)=3\left(3x+1\right)+20x$$

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

$$33x+12-6\left(x+2-\frac{1}{2}x+\frac{2}{6}\right)=3\left(3x+1\right)+20x$$

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

$$33x+12-6\left(x+2-\frac{1}{2}x+\frac{1}{3}\right)=3\left(3x+1\right)+20x$$

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

$$33x+12-6\left(\frac{1}{2}x+2+\frac{1}{3}\right)=3\left(3x+1\right)+20x$$

Convert 2 to fraction \frac{6}{3}.

$$33x+12-6\left(\frac{1}{2}x+\frac{6}{3}+\frac{1}{3}\right)=3\left(3x+1\right)+20x$$

Since \frac{6}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$33x+12-6\left(\frac{1}{2}x+\frac{6+1}{3}\right)=3\left(3x+1\right)+20x$$

Add 6 and 1 to get 7.

$$33x+12-6\left(\frac{1}{2}x+\frac{7}{3}\right)=3\left(3x+1\right)+20x$$

Use the distributive property to multiply -6 by \frac{1}{2}x+\frac{7}{3}.

$$33x+12-6\times \frac{1}{2}x-6\times \frac{7}{3}=3\left(3x+1\right)+20x$$

Multiply -6 and \frac{1}{2} to get \frac{-6}{2}.

$$33x+12+\frac{-6}{2}x-6\times \frac{7}{3}=3\left(3x+1\right)+20x$$

Divide -6 by 2 to get -3.

$$33x+12-3x-6\times \frac{7}{3}=3\left(3x+1\right)+20x$$

Express -6\times \frac{7}{3} as a single fraction.

$$33x+12-3x+\frac{-6\times 7}{3}=3\left(3x+1\right)+20x$$

Multiply -6 and 7 to get -42.

$$33x+12-3x+\frac{-42}{3}=3\left(3x+1\right)+20x$$

Divide -42 by 3 to get -14.

$$33x+12-3x-14=3\left(3x+1\right)+20x$$

Combine 33x and -3x to get 30x.

$$30x+12-14=3\left(3x+1\right)+20x$$

Subtract 14 from 12 to get -2.

$$30x-2=3\left(3x+1\right)+20x$$

Use the distributive property to multiply 3 by 3x+1.

$$30x-2=9x+3+20x$$

Combine 9x and 20x to get 29x.

$$30x-2=29x+3$$

Subtract 29x from both sides.

$$30x-2-29x=3$$

Combine 30x and -29x to get x.

$$x-2=3$$

Add 2 to both sides.

$$x=3+2$$

Add 3 and 2 to get 5.

$$x=5$$