Solve 19-3y^2+3y/2y-2-3x(y-2)=-1

$$\frac { 19 - 3 y ^ { 2 } + 3 y } { 2 y - 2 } - 3 x ( y - 2 ) = - 1$$

Solve for x

$$x=-\frac{17+5y-3y^{2}}{6\left(-y^{2}+3y-2\right)}$$

$$y\neq 1\text{ and }y\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(y-1\right).

$$19-3y^{2}+3y-3x\left(y-2\right)\times 2\left(y-1\right)=-2\left(y-1\right)$$

Multiply 3 and 2 to get 6.

$$19-3y^{2}+3y-6x\left(y-2\right)\left(y-1\right)=-2\left(y-1\right)$$

Use the distributive property to multiply -2 by y-1.

$$19-3y^{2}+3y-6x\left(y-2\right)\left(y-1\right)=-2y+2$$

Use the distributive property to multiply -6x by y-2.

$$19-3y^{2}+3y+\left(-6xy+12x\right)\left(y-1\right)=-2y+2$$

Use the distributive property to multiply -6xy+12x by y-1 and combine like terms.

$$19-3y^{2}+3y-6xy^{2}+18yx-12x=-2y+2$$

Subtract 19 from both sides.

$$-3y^{2}+3y-6xy^{2}+18yx-12x=-2y+2-19$$

Subtract 19 from 2 to get -17.

$$-3y^{2}+3y-6xy^{2}+18yx-12x=-2y-17$$

Add 3y^{2} to both sides.

$$3y-6xy^{2}+18yx-12x=-2y-17+3y^{2}$$

Subtract 3y from both sides.

$$-6xy^{2}+18yx-12x=-2y-17+3y^{2}-3y$$

Combine -2y and -3y to get -5y.

$$-6xy^{2}+18yx-12x=-5y-17+3y^{2}$$

Combine all terms containing x.

$$\left(-6y^{2}+18y-12\right)x=-5y-17+3y^{2}$$

The equation is in standard form.

$$\left(-6y^{2}+18y-12\right)x=3y^{2}-5y-17$$

Divide both sides by -6y^{2}+18y-12.

$$\frac{\left(-6y^{2}+18y-12\right)x}{-6y^{2}+18y-12}=\frac{3y^{2}-5y-17}{-6y^{2}+18y-12}$$

Dividing by -6y^{2}+18y-12 undoes the multiplication by -6y^{2}+18y-12.

$$x=\frac{3y^{2}-5y-17}{-6y^{2}+18y-12}$$

Divide -5y-17+3y^{2} by -6y^{2}+18y-12.

$$x=\frac{3y^{2}-5y-17}{6\left(-y^{2}+3y-2\right)}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-\frac{\sqrt{36x^{2}+444x+229}-18x-5}{6\left(2x+1\right)}\text{; }y=\frac{\sqrt{36x^{2}+444x+229}+18x+5}{6\left(2x+1\right)}\text{, }&x\neq -\frac{1}{2}\\y=\frac{23}{4}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{\sqrt{36x^{2}+444x+229}-18x-5}{6\left(2x+1\right)}\text{; }y=\frac{\sqrt{36x^{2}+444x+229}+18x+5}{6\left(2x+1\right)}\text{, }&x\leq -\frac{\sqrt{285}}{3}-\frac{37}{6}\text{ or }\left(x\neq -\frac{1}{2}\text{ and }x\geq \frac{\sqrt{285}}{3}-\frac{37}{6}\right)\\y=\frac{23}{4}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$