Solve 2x-1/3+3/2=3x-1/z^2

$$\frac { 2 x - 1 } { 3 } + \frac { 3 } { 2 } = \frac { 3 x - 1 } { z ^ { 2 } }$$

Solve for x (complex solution)

$$x=-\frac{7z^{2}+6}{2\left(2z^{2}-9\right)}$$

$$z\neq -\frac{3\sqrt{2}}{2}\text{ and }z\neq \frac{3\sqrt{2}}{2}\text{ and }z\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6z^{2}, the least common multiple of 3,2,z^{2}.

$$2z^{2}\left(2x-1\right)+6z^{2}\times \frac{3}{2}=6\left(3x-1\right)$$

Use the distributive property to multiply 2z^{2} by 2x-1.

$$4z^{2}x-2z^{2}+6z^{2}\times \frac{3}{2}=6\left(3x-1\right)$$

Multiply 6 and \frac{3}{2} to get 9.

$$4z^{2}x-2z^{2}+9z^{2}=6\left(3x-1\right)$$

Combine -2z^{2} and 9z^{2} to get 7z^{2}.

$$4z^{2}x+7z^{2}=6\left(3x-1\right)$$

Use the distributive property to multiply 6 by 3x-1.

$$4z^{2}x+7z^{2}=18x-6$$

Subtract 18x from both sides.

$$4z^{2}x+7z^{2}-18x=-6$$

Subtract 7z^{2} from both sides.

$$4z^{2}x-18x=-6-7z^{2}$$

Combine all terms containing x.

$$\left(4z^{2}-18\right)x=-6-7z^{2}$$

The equation is in standard form.

$$\left(4z^{2}-18\right)x=-7z^{2}-6$$

Divide both sides by 4z^{2}-18.

$$\frac{\left(4z^{2}-18\right)x}{4z^{2}-18}=\frac{-7z^{2}-6}{4z^{2}-18}$$

Dividing by 4z^{2}-18 undoes the multiplication by 4z^{2}-18.

$$x=\frac{-7z^{2}-6}{4z^{2}-18}$$

Divide -6-7z^{2} by 4z^{2}-18.

$$x=-\frac{7z^{2}+6}{2\left(2z^{2}-9\right)}$$

Solve for x

$$x=-\frac{7z^{2}+6}{2\left(2z^{2}-9\right)}$$

$$|z|\neq \frac{3\sqrt{2}}{2}\text{ and }z\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6z^{2}, the least common multiple of 3,2,z^{2}.

$$2z^{2}\left(2x-1\right)+6z^{2}\times \frac{3}{2}=6\left(3x-1\right)$$

Use the distributive property to multiply 2z^{2} by 2x-1.

$$4z^{2}x-2z^{2}+6z^{2}\times \frac{3}{2}=6\left(3x-1\right)$$

Multiply 6 and \frac{3}{2} to get 9.

$$4z^{2}x-2z^{2}+9z^{2}=6\left(3x-1\right)$$

Combine -2z^{2} and 9z^{2} to get 7z^{2}.

$$4z^{2}x+7z^{2}=6\left(3x-1\right)$$

Use the distributive property to multiply 6 by 3x-1.

$$4z^{2}x+7z^{2}=18x-6$$

Subtract 18x from both sides.

$$4z^{2}x+7z^{2}-18x=-6$$

Subtract 7z^{2} from both sides.

$$4z^{2}x-18x=-6-7z^{2}$$

Combine all terms containing x.

$$\left(4z^{2}-18\right)x=-6-7z^{2}$$

The equation is in standard form.

$$\left(4z^{2}-18\right)x=-7z^{2}-6$$

Divide both sides by 4z^{2}-18.

$$\frac{\left(4z^{2}-18\right)x}{4z^{2}-18}=\frac{-7z^{2}-6}{4z^{2}-18}$$

Dividing by 4z^{2}-18 undoes the multiplication by 4z^{2}-18.

$$x=\frac{-7z^{2}-6}{4z^{2}-18}$$

Divide -6-7z^{2} by 4z^{2}-18.

$$x=-\frac{7z^{2}+6}{2\left(2z^{2}-9\right)}$$

Solve for z (complex solution)

$$z=-i\left(4x+7\right)^{-\frac{1}{2}}\sqrt{6-18x}$$

$$z=i\left(4x+7\right)^{-\frac{1}{2}}\sqrt{6-18x}\text{, }x\neq \frac{1}{3}\text{ and }x\neq -\frac{7}{4}$$

Solve for z

$$z=\sqrt{\frac{6\left(3x-1\right)}{4x+7}}$$

$$z=-\sqrt{\frac{6\left(3x-1\right)}{4x+7}}\text{, }x>\frac{1}{3}\text{ or }x<-\frac{7}{4}$$