Solve m/n+9=m-4/m+1

$$\frac { m } { n + 9 } = \frac { m - 4 } { m + 1 }$$

Solve for n

$$n=-\frac{m^{2}-8m+36}{4-m}$$

$$m\neq -1\text{ and }m\neq 0\text{ and }m\neq 4$$

Steps for Solving Linear Equation

Variable n cannot be equal to -9 since division by zero is not defined. Multiply both sides of the equation by \left(m+1\right)\left(n+9\right), the least common multiple of n+9,m+1.

$$\left(m+1\right)m=\left(n+9\right)\left(m-4\right)$$

Use the distributive property to multiply m+1 by m.

$$m^{2}+m=\left(n+9\right)\left(m-4\right)$$

Use the distributive property to multiply n+9 by m-4.

$$m^{2}+m=nm-4n+9m-36$$

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

$$nm-4n+9m-36=m^{2}+m$$

Subtract 9m from both sides.

$$nm-4n-36=m^{2}+m-9m$$

Combine m and -9m to get -8m.

$$nm-4n-36=m^{2}-8m$$

Add 36 to both sides.

$$nm-4n=m^{2}-8m+36$$

Combine all terms containing n.

$$\left(m-4\right)n=m^{2}-8m+36$$

Divide both sides by m-4.

$$\frac{\left(m-4\right)n}{m-4}=\frac{m^{2}-8m+36}{m-4}$$

Dividing by m-4 undoes the multiplication by m-4.

$$n=\frac{m^{2}-8m+36}{m-4}$$

Variable n cannot be equal to -9.

$$n=\frac{m^{2}-8m+36}{m-4}\text{, }n\neq -9$$

Solve for m

$$m=\frac{\sqrt{n^{2}-80}+n+8}{2}$$

$$m=\frac{-\sqrt{n^{2}-80}+n+8}{2}\text{, }n\geq 4\sqrt{5}\text{ or }\left(n\neq -9\text{ and }n\leq -4\sqrt{5}\right)$$