Solve y/2+y+3/y-2=1+8/4-y^2

$$\frac { y } { 2 + y } + \frac { 3 } { y - 2 } = 1 + \frac { 8 } { 4 - y ^ { 2 } }$$

Solve for y

$$y=-18$$

Steps for Solving Linear Equation

Variable y cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(y-2\right)\left(y+2\right), the least common multiple of 2+y,y-2,4-y^{2}.

$$\left(y-2\right)y+\left(y+2\right)\times 3=\left(y-2\right)\left(y+2\right)-8$$

Use the distributive property to multiply y-2 by y.

$$y^{2}-2y+\left(y+2\right)\times 3=\left(y-2\right)\left(y+2\right)-8$$

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

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

Combine -2y and 3y to get y.

$$y^{2}+y+6=\left(y-2\right)\left(y+2\right)-8$$

Consider \left(y-2\right)\left(y+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Subtract 8 from -4 to get -12.

$$y^{2}+y+6=y^{2}-12$$

Subtract y^{2} from both sides.

$$y^{2}+y+6-y^{2}=-12$$

Combine y^{2} and -y^{2} to get 0.

$$y+6=-12$$

Subtract 6 from both sides.

$$y=-12-6$$

Subtract 6 from -12 to get -18.

$$y=-18$$