Solve 2/y+3+3/y-4=5/y+6

$$\frac { 2 } { y + 3 } + \frac { 3 } { y - 4 } = \frac { 5 } { y + 6 }$$

Solve for y

$$y = -\frac{11}{6} = -1\frac{5}{6} \approx -1.833333333$$

Steps for Solving Linear Equation

Variable y cannot be equal to any of the values -6,-3,4 since division by zero is not defined. Multiply both sides of the equation by \left(y-4\right)\left(y+3\right)\left(y+6\right), the least common multiple of y+3,y-4,y+6.

$$\left(y-4\right)\left(y+6\right)\times 2+\left(y+3\right)\left(y+6\right)\times 3=\left(y-4\right)\left(y+3\right)\times 5$$

Use the distributive property to multiply y-4 by y+6 and combine like terms.

$$\left(y^{2}+2y-24\right)\times 2+\left(y+3\right)\left(y+6\right)\times 3=\left(y-4\right)\left(y+3\right)\times 5$$

Use the distributive property to multiply y^{2}+2y-24 by 2.

$$2y^{2}+4y-48+\left(y+3\right)\left(y+6\right)\times 3=\left(y-4\right)\left(y+3\right)\times 5$$

Use the distributive property to multiply y+3 by y+6 and combine like terms.

$$2y^{2}+4y-48+\left(y^{2}+9y+18\right)\times 3=\left(y-4\right)\left(y+3\right)\times 5$$

Use the distributive property to multiply y^{2}+9y+18 by 3.

$$2y^{2}+4y-48+3y^{2}+27y+54=\left(y-4\right)\left(y+3\right)\times 5$$

Combine 2y^{2} and 3y^{2} to get 5y^{2}.

$$5y^{2}+4y-48+27y+54=\left(y-4\right)\left(y+3\right)\times 5$$

Combine 4y and 27y to get 31y.

$$5y^{2}+31y-48+54=\left(y-4\right)\left(y+3\right)\times 5$$

Add -48 and 54 to get 6.

$$5y^{2}+31y+6=\left(y-4\right)\left(y+3\right)\times 5$$

Use the distributive property to multiply y-4 by y+3 and combine like terms.

$$5y^{2}+31y+6=\left(y^{2}-y-12\right)\times 5$$

Use the distributive property to multiply y^{2}-y-12 by 5.

$$5y^{2}+31y+6=5y^{2}-5y-60$$

Subtract 5y^{2} from both sides.

$$5y^{2}+31y+6-5y^{2}=-5y-60$$

Combine 5y^{2} and -5y^{2} to get 0.

$$31y+6=-5y-60$$

Add 5y to both sides.

$$31y+6+5y=-60$$

Combine 31y and 5y to get 36y.

$$36y+6=-60$$

Subtract 6 from both sides.

$$36y=-60-6$$

Subtract 6 from -60 to get -66.

$$36y=-66$$

Divide both sides by 36.

$$y=\frac{-66}{36}$$

Reduce the fraction \frac{-66}{36} to lowest terms by extracting and canceling out 6.

$$y=-\frac{11}{6}$$