Solve y=(a^2-2a-8)x+asigma

$$y = ( a ^ { 2 } - 2 a - 8 ) x + a \sigma$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{a\sigma -y}{\left(a-4\right)\left(a+2\right)}\text{, }&a\neq -2\text{ and }a\neq 4\\x\in \mathrm{C}\text{, }&\left(y=4\sigma \text{ and }a=4\right)\text{ or }\left(y=-2\sigma \text{ and }a=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a^{2}-2a-8 by x.

$$y=a^{2}x-2ax-8x+a\sigma$$

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

$$a^{2}x-2ax-8x+a\sigma =y$$

Subtract a\sigma from both sides.

$$a^{2}x-2ax-8x=y-a\sigma$$

Combine all terms containing x.

$$\left(a^{2}-2a-8\right)x=y-a\sigma$$

Divide both sides by a^{2}-2a-8.

$$\frac{\left(a^{2}-2a-8\right)x}{a^{2}-2a-8}=\frac{y-a\sigma }{a^{2}-2a-8}$$

Dividing by a^{2}-2a-8 undoes the multiplication by a^{2}-2a-8.

$$x=\frac{y-a\sigma }{a^{2}-2a-8}$$

Divide y-a\sigma by a^{2}-2a-8.

$$x=\frac{y-a\sigma }{\left(a-4\right)\left(a+2\right)}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{a\sigma -y}{\left(a-4\right)\left(a+2\right)}\text{, }&a\neq -2\text{ and }a\neq 4\\x\in \mathrm{R}\text{, }&\left(y=4\sigma \text{ and }a=4\right)\text{ or }\left(y=-2\sigma \text{ and }a=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a^{2}-2a-8 by x.

$$y=a^{2}x-2ax-8x+a\sigma$$

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

$$a^{2}x-2ax-8x+a\sigma =y$$

Subtract a\sigma from both sides.

$$a^{2}x-2ax-8x=y-a\sigma$$

Combine all terms containing x.

$$\left(a^{2}-2a-8\right)x=y-a\sigma$$

Divide both sides by a^{2}-2a-8.

$$\frac{\left(a^{2}-2a-8\right)x}{a^{2}-2a-8}=\frac{y-a\sigma }{a^{2}-2a-8}$$

Dividing by a^{2}-2a-8 undoes the multiplication by a^{2}-2a-8.

$$x=\frac{y-a\sigma }{a^{2}-2a-8}$$

Divide y-a\sigma by a^{2}-2a-8.

$$x=\frac{y-a\sigma }{\left(a-4\right)\left(a+2\right)}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{\sqrt{36x^{2}+4xy-4x\sigma +\sigma ^{2}}+\sigma -2x}{2x}\text{; }a=-\frac{-\sqrt{36x^{2}+4xy-4x\sigma +\sigma ^{2}}+\sigma -2x}{2x}\text{, }&x\neq 0\\a=\frac{y}{\sigma }\text{, }&x=0\text{ and }\sigma \neq 0\\a\in \mathrm{C}\text{, }&x=0\text{ and }\sigma =0\text{ and }y=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{\sqrt{36x^{2}+4xy-4x\sigma +\sigma ^{2}}+\sigma -2x}{2x}\text{; }a=-\frac{-\sqrt{36x^{2}+4xy-4x\sigma +\sigma ^{2}}+\sigma -2x}{2x}\text{, }&\left(x>0\text{ or }y\leq -\frac{36x^{2}-4x\sigma +\sigma ^{2}}{4x}\right)\text{ and }\left(y\leq \text{Indeterminate}\text{ or }x\neq 0\right)\text{ and }\left(x<0\text{ or }\left(x\neq 0\text{ and }y\geq -\frac{36x^{2}-4x\sigma +\sigma ^{2}}{4x}\right)\right)\\a=\frac{y}{\sigma }\text{, }&\sigma \neq 0\text{ and }x=0\\a\in \mathrm{R}\text{, }&x=0\text{ and }\sigma =0\text{ and }y=0\end{matrix}\right.$$