Solve ax-1=x-a

$$a x - 1 = x - a$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Add a to both sides.

$$ax-1+a=x$$

Add 1 to both sides.

$$ax+a=x+1$$

Combine all terms containing a.

$$\left(x+1\right)a=x+1$$

Divide both sides by 1+x.

$$\frac{\left(x+1\right)a}{x+1}=\frac{x+1}{x+1}$$

Dividing by 1+x undoes the multiplication by 1+x.

$$a=\frac{x+1}{x+1}$$

Divide 1+x by 1+x.

$$a=1$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-1\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x from both sides.

$$ax-1-x=-a$$

Add 1 to both sides.

$$ax-x=-a+1$$

Combine all terms containing x.

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

The equation is in standard form.

$$\left(a-1\right)x=1-a$$

Divide both sides by -1+a.

$$\frac{\left(a-1\right)x}{a-1}=\frac{1-a}{a-1}$$

Dividing by -1+a undoes the multiplication by -1+a.

$$x=\frac{1-a}{a-1}$$

Divide -a+1 by -1+a.

$$x=-1$$

Solve for a

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Add a to both sides.

$$ax-1+a=x$$

Add 1 to both sides.

$$ax+a=x+1$$

Combine all terms containing a.

$$\left(x+1\right)a=x+1$$

Divide both sides by 1+x.

$$\frac{\left(x+1\right)a}{x+1}=\frac{x+1}{x+1}$$

Dividing by 1+x undoes the multiplication by 1+x.

$$a=\frac{x+1}{x+1}$$

Divide 1+x by 1+x.

$$a=1$$

Solve for x

$$\left\{\begin{matrix}\\x=-1\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x from both sides.

$$ax-1-x=-a$$

Add 1 to both sides.

$$ax-x=-a+1$$

Combine all terms containing x.

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

The equation is in standard form.

$$\left(a-1\right)x=1-a$$

Divide both sides by -1+a.

$$\frac{\left(a-1\right)x}{a-1}=\frac{1-a}{a-1}$$

Dividing by -1+a undoes the multiplication by -1+a.

$$x=\frac{1-a}{a-1}$$

Divide -a+1 by -1+a.

$$x=-1$$