Solve 2(x+1)+y/3(x+1)+1=2x+y/3x+1

$$\frac{ 2(x+1)+y }{ 3(x+1)+1 } = \frac{ 2x+y }{ 3x+1 }$$

Solve for y

$$y = \frac{2}{3} = 0.6666666666666666$$

$$x\neq -\frac{4}{3}\text{ and }x\neq -\frac{1}{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(3x+1\right)\left(3x+4\right), the least common multiple of 3\left(x+1\right)+1,3x+1.

$$\left(3x+1\right)\left(2\left(x+1\right)+y\right)=\left(3x+4\right)\left(2x+y\right)$$

Use the distributive property to multiply 2 by x+1.

$$\left(3x+1\right)\left(2x+2+y\right)=\left(3x+4\right)\left(2x+y\right)$$

Use the distributive property to multiply 3x+1 by 2x+2+y and combine like terms.

$$6x^{2}+8x+3xy+2+y=\left(3x+4\right)\left(2x+y\right)$$

Use the distributive property to multiply 3x+4 by 2x+y.

$$6x^{2}+8x+3xy+2+y=6x^{2}+3xy+8x+4y$$

Subtract 3xy from both sides.

$$6x^{2}+8x+3xy+2+y-3xy=6x^{2}+8x+4y$$

Combine 3xy and -3xy to get 0.

$$6x^{2}+8x+2+y=6x^{2}+8x+4y$$

Subtract 4y from both sides.

$$6x^{2}+8x+2+y-4y=6x^{2}+8x$$

Combine y and -4y to get -3y.

$$6x^{2}+8x+2-3y=6x^{2}+8x$$

Subtract 6x^{2} from both sides.

$$8x+2-3y=6x^{2}+8x-6x^{2}$$

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

$$8x+2-3y=8x$$

Subtract 8x from both sides.

$$2-3y=8x-8x$$

Combine 8x and -8x to get 0.

$$2-3y=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-3y=-2$$

Divide both sides by -3.

$$y=\frac{-2}{-3}$$

Fraction \frac{-2}{-3} can be simplified to \frac{2}{3} by removing the negative sign from both the numerator and the denominator.

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

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus -\frac{4}{3},-\frac{1}{3}$$

$$y = \frac{2}{3} = 0.6666666666666666$$

Solve for x

$$x\in \mathrm{R}\setminus -\frac{4}{3},-\frac{1}{3}$$

$$y = \frac{2}{3} = 0.6666666666666666$$