Solve 2/x-1+kx/x^2-1=3/x+1

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

Solve for k (complex solution)

$$k=\frac{x-5}{x}$$

$$x\neq 0\text{ and }x\neq -1\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Use the distributive property to multiply x+1 by 2.

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

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

$$2x+2+kx=3x-3$$

Subtract 2x from both sides.

$$2+kx=3x-3-2x$$

Combine 3x and -2x to get x.

$$2+kx=x-3$$

Subtract 2 from both sides.

$$kx=x-3-2$$

Subtract 2 from -3 to get -5.

$$kx=x-5$$

The equation is in standard form.

$$xk=x-5$$

Divide both sides by x.

$$\frac{xk}{x}=\frac{x-5}{x}$$

Dividing by x undoes the multiplication by x.

$$k=\frac{x-5}{x}$$

Solve for x (complex solution)

$$x=-\frac{5}{k-1}$$

$$k\neq 6\text{ and }k\neq 1\text{ and }k\neq -4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Use the distributive property to multiply x+1 by 2.

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

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

$$2x+2+kx=3x-3$$

Subtract 3x from both sides.

$$2x+2+kx-3x=-3$$

Combine 2x and -3x to get -x.

$$-x+2+kx=-3$$

Subtract 2 from both sides.

$$-x+kx=-3-2$$

Subtract 2 from -3 to get -5.

$$-x+kx=-5$$

Combine all terms containing x.

$$\left(-1+k\right)x=-5$$

The equation is in standard form.

$$\left(k-1\right)x=-5$$

Divide both sides by -1+k.

$$\frac{\left(k-1\right)x}{k-1}=-\frac{5}{k-1}$$

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

$$x=-\frac{5}{k-1}$$

Variable x cannot be equal to any of the values -1,1.

$$x=-\frac{5}{k-1}\text{, }x\neq -1\text{ and }x\neq 1$$

Solve for k

$$k=\frac{x-5}{x}$$

$$x\neq 0\text{ and }|x|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Use the distributive property to multiply x+1 by 2.

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

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

$$2x+2+kx=3x-3$$

Subtract 2x from both sides.

$$2+kx=3x-3-2x$$

Combine 3x and -2x to get x.

$$2+kx=x-3$$

Subtract 2 from both sides.

$$kx=x-3-2$$

Subtract 2 from -3 to get -5.

$$kx=x-5$$

The equation is in standard form.

$$xk=x-5$$

Divide both sides by x.

$$\frac{xk}{x}=\frac{x-5}{x}$$

Dividing by x undoes the multiplication by x.

$$k=\frac{x-5}{x}$$

Solve for x

$$x=-\frac{5}{k-1}$$

$$k\neq -4\text{ and }k\neq 1\text{ and }k\neq 6$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Use the distributive property to multiply x+1 by 2.

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

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

$$2x+2+kx=3x-3$$

Subtract 3x from both sides.

$$2x+2+kx-3x=-3$$

Combine 2x and -3x to get -x.

$$-x+2+kx=-3$$

Subtract 2 from both sides.

$$-x+kx=-3-2$$

Subtract 2 from -3 to get -5.

$$-x+kx=-5$$

Combine all terms containing x.

$$\left(-1+k\right)x=-5$$

The equation is in standard form.

$$\left(k-1\right)x=-5$$

Divide both sides by -1+k.

$$\frac{\left(k-1\right)x}{k-1}=-\frac{5}{k-1}$$

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

$$x=-\frac{5}{k-1}$$

Variable x cannot be equal to any of the values -1,1.

$$x=-\frac{5}{k-1}\text{, }x\neq -1\text{ and }x\neq 1$$