Solve 6y^2+10y/2y^2-14y=3y+5/y-7

$$\frac { 6 y ^ { 2 } + 10 y } { 2 y ^ { 2 } - 14 y } = \frac { 3 y + 5 } { y - 7 }$$

Solve for y (complex solution)

$$y\in \mathrm{C}\setminus 7,0$$

Solution Steps

Variable y cannot be equal to any of the values 0,7 since division by zero is not defined. Multiply both sides of the equation by 2y\left(y-7\right), the least common multiple of 2y^{2}-14y,y-7.

$$6y^{2}+10y=2y\left(3y+5\right)$$

Use the distributive property to multiply 2y by 3y+5.

$$6y^{2}+10y=6y^{2}+10y$$

Subtract 6y^{2} from both sides.

$$6y^{2}+10y-6y^{2}=10y$$

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

$$10y=10y$$

Subtract 10y from both sides.

$$10y-10y=0$$

Combine 10y and -10y to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Variable y cannot be equal to any of the values 7,0.

$$y\in \mathrm{C}\setminus 0,7$$

Solve for y

$$y\in \mathrm{R}\setminus 7,0$$

Solution Steps

Variable y cannot be equal to any of the values 0,7 since division by zero is not defined. Multiply both sides of the equation by 2y\left(y-7\right), the least common multiple of 2y^{2}-14y,y-7.

$$6y^{2}+10y=2y\left(3y+5\right)$$

Use the distributive property to multiply 2y by 3y+5.

$$6y^{2}+10y=6y^{2}+10y$$

Subtract 6y^{2} from both sides.

$$6y^{2}+10y-6y^{2}=10y$$

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

$$10y=10y$$

Subtract 10y from both sides.

$$10y-10y=0$$

Combine 10y and -10y to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$

Variable y cannot be equal to any of the values 7,0.

$$y\in \mathrm{R}\setminus 0,7$$