Solve z=6(x-y)-3x^2-3y

$$z = 6 ( x - y ) - 3 x ^ { 2 } - 3 y$$

Solve for y

$$y=-\frac{x^{2}}{3}+\frac{2x}{3}-\frac{z}{9}$$

Steps for Solving Linear Equation

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

$$z=6x-6y-3x^{2}-3y$$

Combine -6y and -3y to get -9y.

$$z=6x-9y-3x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$6x-9y-3x^{2}=z$$

Subtract 6x from both sides.

$$-9y-3x^{2}=z-6x$$

Add 3x^{2} to both sides.

$$-9y=z-6x+3x^{2}$$

The equation is in standard form.

$$-9y=3x^{2}-6x+z$$

Divide both sides by -9.

$$\frac{-9y}{-9}=\frac{3x^{2}-6x+z}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$y=\frac{3x^{2}-6x+z}{-9}$$

Divide z-6x+3x^{2} by -9.

$$y=-\frac{x^{2}}{3}+\frac{2x}{3}-\frac{z}{9}$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{9-3z-27y}}{3}+1$$

$$x=\frac{\sqrt{9-3z-27y}}{3}+1$$

Solve for x

$$x=-\frac{\sqrt{9-3z-27y}}{3}+1$$

$$x=\frac{\sqrt{9-3z-27y}}{3}+1\text{, }z\leq 3-9y$$