Solve y=(ax-e^x)(x-2)

$$y = ( a x - e ^ { x } ) ( x - 2 )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{xe^{x}+y-2e^{x}}{x\left(2-x\right)}\text{, }&x\neq 2\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&\left(y=2\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax-e^{x} by x-2.

$$y=\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)$$

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

$$\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)=y$$

Use the distributive property to multiply ax-e^{x} by x.

$$ax^{2}-e^{x}x-2\left(ax-e^{x}\right)=y$$

Use the distributive property to multiply -2 by ax-e^{x}.

$$ax^{2}-e^{x}x-2ax+2e^{x}=y$$

Add e^{x}x to both sides.

$$ax^{2}-2ax+2e^{x}=y+e^{x}x$$

Subtract 2e^{x} from both sides.

$$ax^{2}-2ax=y+e^{x}x-2e^{x}$$

Combine all terms containing a.

$$\left(x^{2}-2x\right)a=y+e^{x}x-2e^{x}$$

The equation is in standard form.

$$\left(x^{2}-2x\right)a=xe^{x}+y-2e^{x}$$

Divide both sides by x^{2}-2x.

$$\frac{\left(x^{2}-2x\right)a}{x^{2}-2x}=\frac{xe^{x}+y-2e^{x}}{x^{2}-2x}$$

Dividing by x^{2}-2x undoes the multiplication by x^{2}-2x.

$$a=\frac{xe^{x}+y-2e^{x}}{x^{2}-2x}$$

Divide y+e^{x}x-2e^{x} by x^{2}-2x.

$$a=\frac{xe^{x}+y-2e^{x}}{x\left(x-2\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{xe^{x}+y-2e^{x}}{x\left(2-x\right)}\text{, }&x\neq 2\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&\left(y=2\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax-e^{x} by x-2.

$$y=\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)$$

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

$$\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)=y$$

Use the distributive property to multiply ax-e^{x} by x.

$$ax^{2}-e^{x}x-2\left(ax-e^{x}\right)=y$$

Use the distributive property to multiply -2 by ax-e^{x}.

$$ax^{2}-e^{x}x-2ax+2e^{x}=y$$

Add e^{x}x to both sides.

$$ax^{2}-2ax+2e^{x}=y+e^{x}x$$

Subtract 2e^{x} from both sides.

$$ax^{2}-2ax=y+e^{x}x-2e^{x}$$

Combine all terms containing a.

$$\left(x^{2}-2x\right)a=y+e^{x}x-2e^{x}$$

The equation is in standard form.

$$\left(x^{2}-2x\right)a=xe^{x}+y-2e^{x}$$

Divide both sides by x^{2}-2x.

$$\frac{\left(x^{2}-2x\right)a}{x^{2}-2x}=\frac{xe^{x}+y-2e^{x}}{x^{2}-2x}$$

Dividing by x^{2}-2x undoes the multiplication by x^{2}-2x.

$$a=\frac{xe^{x}+y-2e^{x}}{x^{2}-2x}$$

Divide y+e^{x}x-2e^{x} by x^{2}-2x.

$$a=\frac{xe^{x}+y-2e^{x}}{x\left(x-2\right)}$$