Solve x^2-2x-8/x^2-11x+28:x^2-6x+5/x^2-12x+35=x-m/x+n

$$\frac { x ^ { 2 } - 2 x - 8 } { x ^ { 2 } - 11 x + 28 } : \frac { x ^ { 2 } - 6 x + 5 } { x ^ { 2 } - 12 x + 35 } = \frac { x - m } { x + n }$$

Solve for m (complex solution)

$$m=-\frac{nx+3x+2n}{x-1}$$

$$x\neq 4\text{ and }x\neq 7\text{ and }x\neq 1\text{ and }x\neq 5\text{ and }x\neq -n$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+n.

$$\left(x+n\right)\left(-1+x\right)^{-1}\left(-7+x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Use the distributive property to multiply x+n by \left(-1+x\right)^{-1}.

$$\left(x\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}\right)\left(-7+x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Use the distributive property to multiply x\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1} by -7+x.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Factor the expressions that are not already factored in \frac{x^{2}-2x-8}{x^{2}-11x+28}.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{\left(x-4\right)\left(x+2\right)}{\left(x-7\right)\left(x-4\right)}=x-m$$

Cancel out x-4 in both numerator and denominator.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x+2}{x-7}=x-m$$

Express \left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x+2}{x-7} as a single fraction.

$$\frac{\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\left(x+2\right)}{x-7}=x-m$$

Factor the expressions that are not already factored in \frac{\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\left(x+2\right)}{x-7}.

$$\frac{\frac{1}{x-1}\left(x-7\right)\left(x+2\right)\left(x+n\right)}{x-7}=x-m$$

Cancel out x-7 in both numerator and denominator.

$$\frac{1}{x-1}\left(x+2\right)\left(x+n\right)=x-m$$

Expand the expression.

$$\frac{1}{x-1}nx+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x=x-m$$

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

$$x-m=\frac{1}{x-1}nx+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{1}{x-1}n as a single fraction.

$$x-m=\frac{n}{x-1}x+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{n}{x-1}x as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{1}{x-1}x^{2} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{2}{x-1}n as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+2\times \frac{1}{x-1}x$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2}{x-1}x$$

Express \frac{2}{x-1}x as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx}{x-1} and \frac{x^{2}}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx+x^{2}}{x-1} and \frac{2n}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}+2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx+x^{2}+2n}{x-1} and \frac{2x}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}+2n+2x}{x-1}$$

Combine like terms in nx+x^{2}+2n+2x.

$$x-m=\frac{2x+2n+x^{2}+xn}{x-1}$$

Subtract x from both sides.

$$-m=\frac{2x+2n+x^{2}+xn}{x-1}-x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-1}{x-1}.

$$-m=\frac{2x+2n+x^{2}+xn}{x-1}-\frac{x\left(x-1\right)}{x-1}$$

Since \frac{2x+2n+x^{2}+xn}{x-1} and \frac{x\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$-m=\frac{2x+2n+x^{2}+xn-x\left(x-1\right)}{x-1}$$

Do the multiplications in 2x+2n+x^{2}+xn-x\left(x-1\right).

$$-m=\frac{2x+2n+x^{2}+xn-x^{2}+x}{x-1}$$

Combine like terms in 2x+2n+x^{2}+xn-x^{2}+x.

$$-m=\frac{3x+2n+xn}{x-1}$$

Multiply both sides of the equation by x-1.

$$-m\left(x-1\right)=3x+2n+xn$$

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

$$-mx+m=3x+2n+xn$$

Combine all terms containing m.

$$\left(-x+1\right)m=3x+2n+xn$$

The equation is in standard form.

$$\left(1-x\right)m=nx+3x+2n$$

Divide both sides by 1-x.

$$\frac{\left(1-x\right)m}{1-x}=\frac{nx+3x+2n}{1-x}$$

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

$$m=\frac{nx+3x+2n}{1-x}$$

Solve for m

$$m=-\frac{nx+3x+2n}{x-1}$$

$$x\neq 7\text{ and }x\neq 4\text{ and }x\neq 1\text{ and }x\neq 5\text{ and }x\neq -n$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+n.

$$\left(x+n\right)\left(-1+x\right)^{-1}\left(-7+x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Use the distributive property to multiply x+n by \left(-1+x\right)^{-1}.

$$\left(x\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}\right)\left(-7+x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Use the distributive property to multiply x\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1} by -7+x.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x^{2}-2x-8}{x^{2}-11x+28}=x-m$$

Factor the expressions that are not already factored in \frac{x^{2}-2x-8}{x^{2}-11x+28}.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{\left(x-4\right)\left(x+2\right)}{\left(x-7\right)\left(x-4\right)}=x-m$$

Cancel out x-4 in both numerator and denominator.

$$\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x+2}{x-7}=x-m$$

Express \left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\times \frac{x+2}{x-7} as a single fraction.

$$\frac{\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\left(x+2\right)}{x-7}=x-m$$

Factor the expressions that are not already factored in \frac{\left(-7x\left(-1+x\right)^{-1}+\left(-1+x\right)^{-1}x^{2}-7n\left(-1+x\right)^{-1}+n\left(-1+x\right)^{-1}x\right)\left(x+2\right)}{x-7}.

$$\frac{\frac{1}{x-1}\left(x-7\right)\left(x+2\right)\left(x+n\right)}{x-7}=x-m$$

Cancel out x-7 in both numerator and denominator.

$$\frac{1}{x-1}\left(x+2\right)\left(x+n\right)=x-m$$

Expand the expression.

$$\frac{1}{x-1}nx+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x=x-m$$

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

$$x-m=\frac{1}{x-1}nx+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{1}{x-1}n as a single fraction.

$$x-m=\frac{n}{x-1}x+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{n}{x-1}x as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{1}{x-1}x^{2}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{1}{x-1}x^{2} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+2\times \frac{1}{x-1}n+2\times \frac{1}{x-1}x$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2}{x-1}n+2\times \frac{1}{x-1}x$$

Express \frac{2}{x-1}n as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+2\times \frac{1}{x-1}x$$

Express 2\times \frac{1}{x-1} as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2}{x-1}x$$

Express \frac{2}{x-1}x as a single fraction.

$$x-m=\frac{nx}{x-1}+\frac{x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx}{x-1} and \frac{x^{2}}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}}{x-1}+\frac{2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx+x^{2}}{x-1} and \frac{2n}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}+2n}{x-1}+\frac{2x}{x-1}$$

Since \frac{nx+x^{2}+2n}{x-1} and \frac{2x}{x-1} have the same denominator, add them by adding their numerators.

$$x-m=\frac{nx+x^{2}+2n+2x}{x-1}$$

Combine like terms in nx+x^{2}+2n+2x.

$$x-m=\frac{2x+2n+x^{2}+xn}{x-1}$$

Subtract x from both sides.

$$-m=\frac{2x+2n+x^{2}+xn}{x-1}-x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-1}{x-1}.

$$-m=\frac{2x+2n+x^{2}+xn}{x-1}-\frac{x\left(x-1\right)}{x-1}$$

Since \frac{2x+2n+x^{2}+xn}{x-1} and \frac{x\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$-m=\frac{2x+2n+x^{2}+xn-x\left(x-1\right)}{x-1}$$

Do the multiplications in 2x+2n+x^{2}+xn-x\left(x-1\right).

$$-m=\frac{2x+2n+x^{2}+xn-x^{2}+x}{x-1}$$

Combine like terms in 2x+2n+x^{2}+xn-x^{2}+x.

$$-m=\frac{3x+2n+xn}{x-1}$$

Multiply both sides of the equation by x-1.

$$-m\left(x-1\right)=3x+2n+xn$$

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

$$-mx+m=3x+2n+xn$$

Combine all terms containing m.

$$\left(-x+1\right)m=3x+2n+xn$$

The equation is in standard form.

$$\left(1-x\right)m=nx+3x+2n$$

Divide both sides by 1-x.

$$\frac{\left(1-x\right)m}{1-x}=\frac{nx+3x+2n}{1-x}$$

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

$$m=\frac{nx+3x+2n}{1-x}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{mx+3x-m}{x+2}\text{, }&x\neq 4\text{ and }x\neq 7\text{ and }x\neq 1\text{ and }x\neq 5\text{ and }x\neq m\text{ and }x\neq -2\\n\neq 2\text{, }&m=-2\text{ and }x=-2\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{mx+3x-m}{x+2}\text{, }&x\neq 7\text{ and }x\neq 4\text{ and }x\neq 1\text{ and }x\neq 5\text{ and }x\neq m\text{ and }x\neq -2\\n\neq 2\text{, }&m=-2\text{ and }x=-2\end{matrix}\right.$$