Solve x^2+(2m-1)x-2m=(k-1)x+2m-k+2

$$x ^ { 2 } + ( 2 m - 1 ) x - 2 m = ( k - 1 ) x + 2 m - k + 2$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=-\frac{2+4m-2mx-x^{2}}{x-1}\text{, }&x\neq 1\\k\in \mathrm{C}\text{, }&x=1\text{ and }m=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m-1 by x.

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

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

$$x^{2}+2mx-x-2m=kx-x+2m-k+2$$

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

$$kx-x+2m-k+2=x^{2}+2mx-x-2m$$

Add x to both sides.

$$kx+2m-k+2=x^{2}+2mx-x-2m+x$$

Combine -x and x to get 0.

$$kx+2m-k+2=x^{2}+2mx-2m$$

Subtract 2m from both sides.

$$kx-k+2=x^{2}+2mx-2m-2m$$

Combine -2m and -2m to get -4m.

$$kx-k+2=x^{2}+2mx-4m$$

Subtract 2 from both sides.

$$kx-k=x^{2}+2mx-4m-2$$

Combine all terms containing k.

$$\left(x-1\right)k=x^{2}+2mx-4m-2$$

Divide both sides by -1+x.

$$\frac{\left(x-1\right)k}{x-1}=\frac{x^{2}+2mx-4m-2}{x-1}$$

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

$$k=\frac{x^{2}+2mx-4m-2}{x-1}$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{x^{2}-kx+k-2}{2\left(x-2\right)}\text{, }&x\neq 2\\m\in \mathrm{C}\text{, }&x=2\text{ and }k=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m-1 by x.

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

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

$$x^{2}+2mx-x-2m=kx-x+2m-k+2$$

Subtract 2m from both sides.

$$x^{2}+2mx-x-2m-2m=kx-x-k+2$$

Combine -2m and -2m to get -4m.

$$x^{2}+2mx-x-4m=kx-x-k+2$$

Subtract x^{2} from both sides.

$$2mx-x-4m=kx-x-k+2-x^{2}$$

Add x to both sides.

$$2mx-4m=kx-x-k+2-x^{2}+x$$

Combine -x and x to get 0.

$$2mx-4m=kx-k+2-x^{2}$$

Combine all terms containing m.

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

The equation is in standard form.

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

Divide both sides by -4+2x.

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

Dividing by -4+2x undoes the multiplication by -4+2x.

$$m=\frac{2-k+kx-x^{2}}{2x-4}$$

Divide kx-k+2-x^{2} by -4+2x.

$$m=\frac{2-k+kx-x^{2}}{2\left(x-2\right)}$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{2+4m-2mx-x^{2}}{x-1}\text{, }&x\neq 1\\k\in \mathrm{R}\text{, }&x=1\text{ and }m=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m-1 by x.

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

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

$$x^{2}+2mx-x-2m=kx-x+2m-k+2$$

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

$$kx-x+2m-k+2=x^{2}+2mx-x-2m$$

Add x to both sides.

$$kx+2m-k+2=x^{2}+2mx-x-2m+x$$

Combine -x and x to get 0.

$$kx+2m-k+2=x^{2}+2mx-2m$$

Subtract 2m from both sides.

$$kx-k+2=x^{2}+2mx-2m-2m$$

Combine -2m and -2m to get -4m.

$$kx-k+2=x^{2}+2mx-4m$$

Subtract 2 from both sides.

$$kx-k=x^{2}+2mx-4m-2$$

Combine all terms containing k.

$$\left(x-1\right)k=x^{2}+2mx-4m-2$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)k}{x-1}=\frac{x^{2}+2mx-4m-2}{x-1}$$

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

$$k=\frac{x^{2}+2mx-4m-2}{x-1}$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{x^{2}-kx+k-2}{2\left(x-2\right)}\text{, }&x\neq 2\\m\in \mathrm{R}\text{, }&x=2\text{ and }k=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m-1 by x.

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

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

$$x^{2}+2mx-x-2m=kx-x+2m-k+2$$

Subtract 2m from both sides.

$$x^{2}+2mx-x-2m-2m=kx-x-k+2$$

Combine -2m and -2m to get -4m.

$$x^{2}+2mx-x-4m=kx-x-k+2$$

Subtract x^{2} from both sides.

$$2mx-x-4m=kx-x-k+2-x^{2}$$

Add x to both sides.

$$2mx-4m=kx-x-k+2-x^{2}+x$$

Combine -x and x to get 0.

$$2mx-4m=kx-k+2-x^{2}$$

Combine all terms containing m.

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

The equation is in standard form.

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

Divide both sides by -4+2x.

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

Dividing by -4+2x undoes the multiplication by -4+2x.

$$m=\frac{2-k+kx-x^{2}}{2x-4}$$

Divide kx-k+2-x^{2} by -4+2x.

$$m=\frac{2-k+kx-x^{2}}{2\left(x-2\right)}$$