Solve ax-1/a+3-x-1/2-a=3(x-1)/a^2+a-6

$$\frac { a x - 1 } { a + 3 } - \frac { x - 1 } { 2 - a } = \frac { 3 ( x - 1 ) } { a ^ { 2 } + a - 6 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{2}{a}\text{, }&a\neq 0\text{ and }a\neq -3\text{ and }a\neq 2\\x\in \mathrm{C}\text{, }&a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-2\right)\left(a+3\right), the least common multiple of a+3,2-a,a^{2}+a-6.

$$\left(a-2\right)\left(ax-1\right)-\left(-\left(3+a\right)\left(x-1\right)\right)=3\left(x-1\right)$$

Use the distributive property to multiply a-2 by ax-1.

$$xa^{2}-a-2ax+2-\left(-\left(3+a\right)\left(x-1\right)\right)=3\left(x-1\right)$$

Use the distributive property to multiply -1 by 3+a.

$$xa^{2}-a-2ax+2-\left(-3-a\right)\left(x-1\right)=3\left(x-1\right)$$

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

$$xa^{2}-a-2ax+2-\left(-3x+3-ax+a\right)=3\left(x-1\right)$$

To find the opposite of -3x+3-ax+a, find the opposite of each term.

$$xa^{2}-a-2ax+2+3x-3+ax-a=3\left(x-1\right)$$

Subtract 3 from 2 to get -1.

$$xa^{2}-a-2ax-1+3x+ax-a=3\left(x-1\right)$$

Combine -2ax and ax to get -ax.

$$xa^{2}-a-ax-1+3x-a=3\left(x-1\right)$$

Combine -a and -a to get -2a.

$$xa^{2}-2a-ax-1+3x=3\left(x-1\right)$$

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

$$xa^{2}-2a-ax-1+3x=3x-3$$

Subtract 3x from both sides.

$$xa^{2}-2a-ax-1+3x-3x=-3$$

Combine 3x and -3x to get 0.

$$xa^{2}-2a-ax-1=-3$$

Add 2a to both sides.

$$xa^{2}-ax-1=-3+2a$$

Add 1 to both sides.

$$xa^{2}-ax=-3+2a+1$$

Add -3 and 1 to get -2.

$$xa^{2}-ax=-2+2a$$

Combine all terms containing x.

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

The equation is in standard form.

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

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

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

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

$$x=\frac{2a-2}{a^{2}-a}$$

Divide -2+2a by a^{2}-a.

$$x=\frac{2}{a}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{2}{a}\text{, }&a\neq 0\text{ and }a\neq -3\text{ and }a\neq 2\\x\in \mathrm{R}\text{, }&a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-2\right)\left(a+3\right), the least common multiple of a+3,2-a,a^{2}+a-6.

$$\left(a-2\right)\left(ax-1\right)-\left(-\left(3+a\right)\left(x-1\right)\right)=3\left(x-1\right)$$

Use the distributive property to multiply a-2 by ax-1.

$$xa^{2}-a-2ax+2-\left(-\left(3+a\right)\left(x-1\right)\right)=3\left(x-1\right)$$

Use the distributive property to multiply -1 by 3+a.

$$xa^{2}-a-2ax+2-\left(-3-a\right)\left(x-1\right)=3\left(x-1\right)$$

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

$$xa^{2}-a-2ax+2-\left(-3x+3-ax+a\right)=3\left(x-1\right)$$

To find the opposite of -3x+3-ax+a, find the opposite of each term.

$$xa^{2}-a-2ax+2+3x-3+ax-a=3\left(x-1\right)$$

Subtract 3 from 2 to get -1.

$$xa^{2}-a-2ax-1+3x+ax-a=3\left(x-1\right)$$

Combine -2ax and ax to get -ax.

$$xa^{2}-a-ax-1+3x-a=3\left(x-1\right)$$

Combine -a and -a to get -2a.

$$xa^{2}-2a-ax-1+3x=3\left(x-1\right)$$

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

$$xa^{2}-2a-ax-1+3x=3x-3$$

Subtract 3x from both sides.

$$xa^{2}-2a-ax-1+3x-3x=-3$$

Combine 3x and -3x to get 0.

$$xa^{2}-2a-ax-1=-3$$

Add 2a to both sides.

$$xa^{2}-ax-1=-3+2a$$

Add 1 to both sides.

$$xa^{2}-ax=-3+2a+1$$

Add -3 and 1 to get -2.

$$xa^{2}-ax=-2+2a$$

Combine all terms containing x.

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

The equation is in standard form.

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

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

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

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

$$x=\frac{2a-2}{a^{2}-a}$$

Divide -2+2a by a^{2}-a.

$$x=\frac{2}{a}$$

Solve for a

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a=\frac{2}{x}\text{, }&x\neq -\frac{2}{3}\text{ and }x\neq 0\text{ and }x\neq 1\end{matrix}\right.$$