Solve z^3-5z^2/z^2+5z-50=-19z-9n/z+10

$$\frac { z ^ { 3 } - 5 z ^ { 2 } } { z ^ { 2 } + 5 z - 50 } = \frac { - 19 z - 9 n } { z + 10 }$$

Solve for n

$$n=-\frac{z\left(z+19\right)}{9}$$

$$z\neq 5\text{ and }z\neq -10$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(z-5\right)\left(z+10\right), the least common multiple of z^{2}+5z-50,z+10.

$$z^{3}-5z^{2}=\left(z-5\right)\left(-19z-9n\right)$$

Use the distributive property to multiply z-5 by -19z-9n.

$$z^{3}-5z^{2}=-19z^{2}-9zn+95z+45n$$

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

$$-19z^{2}-9zn+95z+45n=z^{3}-5z^{2}$$

Add 19z^{2} to both sides.

$$-9zn+95z+45n=z^{3}-5z^{2}+19z^{2}$$

Combine -5z^{2} and 19z^{2} to get 14z^{2}.

$$-9zn+95z+45n=z^{3}+14z^{2}$$

Subtract 95z from both sides.

$$-9zn+45n=z^{3}+14z^{2}-95z$$

Combine all terms containing n.

$$\left(-9z+45\right)n=z^{3}+14z^{2}-95z$$

The equation is in standard form.

$$\left(45-9z\right)n=z^{3}+14z^{2}-95z$$

Divide both sides by -9z+45.

$$\frac{\left(45-9z\right)n}{45-9z}=\frac{z\left(z-5\right)\left(z+19\right)}{45-9z}$$

Dividing by -9z+45 undoes the multiplication by -9z+45.

$$n=\frac{z\left(z-5\right)\left(z+19\right)}{45-9z}$$

Divide z\left(-5+z\right)\left(19+z\right) by -9z+45.

$$n=-\frac{z\left(z+19\right)}{9}$$

Solve for z

$$\left\{\begin{matrix}z=\frac{-\sqrt{361-36n}-19}{2}\text{, }&n\neq 10\text{ and }n\leq \frac{361}{36}\\z=\frac{\sqrt{361-36n}-19}{2}\text{, }&n\neq -\frac{40}{3}\text{ and }n\leq \frac{361}{36}\end{matrix}\right.$$