Solve x^3+12x/x^3+12x=y^3+27y/y^3+27y

$$\frac { x ^ { 3 } + 12 x } { x ^ { 3 } + 12 x } = \frac { y ^ { 3 } + 27 y } { y ^ { 3 } + 27 y }$$

Solve for x

$$x\neq 0$$

$$y\neq 0$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xy\left(x^{2}+12\right)\left(y^{2}+27\right), the least common multiple of x^{3}+12x,y^{3}+27y.

$$y\left(y^{2}+27\right)\left(x^{3}+12x\right)=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply y by y^{2}+27.

$$\left(y^{3}+27y\right)\left(x^{3}+12x\right)=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply y^{3}+27y by x^{3}+12x.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply x by x^{2}+12.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=\left(x^{3}+12x\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply x^{3}+12x by y^{3}+27y.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=x^{3}y^{3}+27x^{3}y+12xy^{3}+324yx$$

Subtract x^{3}y^{3} from both sides.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx-x^{3}y^{3}=27x^{3}y+12xy^{3}+324yx$$

Combine y^{3}x^{3} and -x^{3}y^{3} to get 0.

$$12y^{3}x+27yx^{3}+324yx=27x^{3}y+12xy^{3}+324yx$$

Subtract 27x^{3}y from both sides.

$$12y^{3}x+27yx^{3}+324yx-27x^{3}y=12xy^{3}+324yx$$

Combine 27yx^{3} and -27x^{3}y to get 0.

$$12y^{3}x+324yx=12xy^{3}+324yx$$

Subtract 12xy^{3} from both sides.

$$12y^{3}x+324yx-12xy^{3}=324yx$$

Combine 12y^{3}x and -12xy^{3} to get 0.

$$324yx=324yx$$

Subtract 324yx from both sides.

$$324yx-324yx=0$$

Combine 324yx and -324yx to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to 0.

$$x\in \mathrm{R}\setminus 0$$

Solve for y

$$y\neq 0$$

$$x\neq 0$$

Solution Steps

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xy\left(x^{2}+12\right)\left(y^{2}+27\right), the least common multiple of x^{3}+12x,y^{3}+27y.

$$y\left(y^{2}+27\right)\left(x^{3}+12x\right)=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply y by y^{2}+27.

$$\left(y^{3}+27y\right)\left(x^{3}+12x\right)=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply y^{3}+27y by x^{3}+12x.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=x\left(x^{2}+12\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply x by x^{2}+12.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=\left(x^{3}+12x\right)\left(y^{3}+27y\right)$$

Use the distributive property to multiply x^{3}+12x by y^{3}+27y.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx=x^{3}y^{3}+27x^{3}y+12xy^{3}+324yx$$

Subtract x^{3}y^{3} from both sides.

$$y^{3}x^{3}+12y^{3}x+27yx^{3}+324yx-x^{3}y^{3}=27x^{3}y+12xy^{3}+324yx$$

Combine y^{3}x^{3} and -x^{3}y^{3} to get 0.

$$12y^{3}x+27yx^{3}+324yx=27x^{3}y+12xy^{3}+324yx$$

Subtract 27x^{3}y from both sides.

$$12y^{3}x+27yx^{3}+324yx-27x^{3}y=12xy^{3}+324yx$$

Combine 27yx^{3} and -27x^{3}y to get 0.

$$12y^{3}x+324yx=12xy^{3}+324yx$$

Subtract 12xy^{3} from both sides.

$$12y^{3}x+324yx-12xy^{3}=324yx$$

Combine 12y^{3}x and -12xy^{3} to get 0.

$$324yx=324yx$$

Subtract 324yx from both sides.

$$324yx-324yx=0$$

Combine 324yx and -324yx 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 0.

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