Solve x/10+(2-3x)^2/30+x/10(1-x)+2/15(1+5x)=frac{x}{5}(3+x)-frac{x-2}{6}

$$\frac { x } { 10 } + \frac { ( 2 - 3 x ) ^ { 2 } } { 30 } + \frac { x } { 10 } ( 1 - x ) + \frac { 2 } { 15 } ( 1 + 5 x ) = \frac { x } { 5 } ( 3 + x ) - \frac { x - 2 } { 6 }$$

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 10,30,15,5,6.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-3x\right)^{2}.

$$3x+4-12x+9x^{2}+3x\left(1-x\right)+4\left(1+5x\right)=6x\left(3+x\right)-5\left(x-2\right)$$

Combine 3x and -12x to get -9x.

$$-9x+4+9x^{2}+3x\left(1-x\right)+4\left(1+5x\right)=6x\left(3+x\right)-5\left(x-2\right)$$

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

$$-9x+4+9x^{2}+3x-3x^{2}+4\left(1+5x\right)=6x\left(3+x\right)-5\left(x-2\right)$$

Combine -9x and 3x to get -6x.

$$-6x+4+9x^{2}-3x^{2}+4\left(1+5x\right)=6x\left(3+x\right)-5\left(x-2\right)$$

Combine 9x^{2} and -3x^{2} to get 6x^{2}.

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

Use the distributive property to multiply 4 by 1+5x.

$$-6x+4+6x^{2}+4+20x=6x\left(3+x\right)-5\left(x-2\right)$$

Add 4 and 4 to get 8.

$$-6x+8+6x^{2}+20x=6x\left(3+x\right)-5\left(x-2\right)$$

Combine -6x and 20x to get 14x.

$$14x+8+6x^{2}=6x\left(3+x\right)-5\left(x-2\right)$$

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

$$14x+8+6x^{2}=18x+6x^{2}-5\left(x-2\right)$$

Use the distributive property to multiply -5 by x-2.

$$14x+8+6x^{2}=18x+6x^{2}-5x+10$$

Combine 18x and -5x to get 13x.

$$14x+8+6x^{2}=13x+6x^{2}+10$$

Subtract 13x from both sides.

$$14x+8+6x^{2}-13x=6x^{2}+10$$

Combine 14x and -13x to get x.

$$x+8+6x^{2}=6x^{2}+10$$

Subtract 6x^{2} from both sides.

$$x+8+6x^{2}-6x^{2}=10$$

Combine 6x^{2} and -6x^{2} to get 0.

$$x+8=10$$

Subtract 8 from both sides.

$$x=10-8$$

Subtract 8 from 10 to get 2.

$$x=2$$