Solve x/x-3-2=k/x-1

$$\frac { x } { x - 3 } - 2 = \frac { k } { x - 1 }$$

Solve for k

$$k=-\frac{\left(1-x\right)\left(x-6\right)}{3-x}$$

$$x\neq 3\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-3\right)\left(x-1\right), the least common multiple of x-3,x-1.

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

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

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

Use the distributive property to multiply x-3 by x-1 and combine like terms.

$$x^{2}-x+\left(x^{2}-4x+3\right)\left(-2\right)=\left(x-3\right)k$$

Use the distributive property to multiply x^{2}-4x+3 by -2.

$$x^{2}-x-2x^{2}+8x-6=\left(x-3\right)k$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-x+8x-6=\left(x-3\right)k$$

Combine -x and 8x to get 7x.

$$-x^{2}+7x-6=\left(x-3\right)k$$

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

$$-x^{2}+7x-6=xk-3k$$

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

$$xk-3k=-x^{2}+7x-6$$

Combine all terms containing k.

$$\left(x-3\right)k=-x^{2}+7x-6$$

Divide both sides by x-3.

$$\frac{\left(x-3\right)k}{x-3}=\frac{\left(1-x\right)\left(x-6\right)}{x-3}$$

Dividing by x-3 undoes the multiplication by x-3.

$$k=\frac{\left(1-x\right)\left(x-6\right)}{x-3}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt{k^{2}-2k+25}-k+7}{2}\text{, }&\text{unconditionally}\\x=\frac{-\sqrt{k^{2}-2k+25}-k+7}{2}\text{, }&k\neq 0\end{matrix}\right.$$