Solve (nY-1)+4Y=-40/n-6

$$( n Y - 1 ) + 4 Y = \frac { - 40 } { n - 6 }$$

Solve for Y

$$Y=-\frac{46-n}{\left(n-6\right)\left(n+4\right)}$$

$$n\neq -4\text{ and }n\neq 6$$

Steps for Solving Linear Equation

Multiply both sides of the equation by n-6.

$$nY\left(n-6\right)+\left(n-6\right)\left(-1\right)+4Y\left(n-6\right)=-40$$

Use the distributive property to multiply nY by n-6.

$$Yn^{2}-6nY+\left(n-6\right)\left(-1\right)+4Y\left(n-6\right)=-40$$

Use the distributive property to multiply n-6 by -1.

$$Yn^{2}-6nY-n+6+4Y\left(n-6\right)=-40$$

Use the distributive property to multiply 4Y by n-6.

$$Yn^{2}-6nY-n+6+4Yn-24Y=-40$$

Combine -6nY and 4Yn to get -2nY.

$$Yn^{2}-2nY-n+6-24Y=-40$$

Add n to both sides.

$$Yn^{2}-2nY+6-24Y=-40+n$$

Subtract 6 from both sides.

$$Yn^{2}-2nY-24Y=-40+n-6$$

Subtract 6 from -40 to get -46.

$$Yn^{2}-2nY-24Y=-46+n$$

Combine all terms containing Y.

$$\left(n^{2}-2n-24\right)Y=-46+n$$

The equation is in standard form.

$$\left(n^{2}-2n-24\right)Y=n-46$$

Divide both sides by n^{2}-2n-24.

$$\frac{\left(n^{2}-2n-24\right)Y}{n^{2}-2n-24}=\frac{n-46}{n^{2}-2n-24}$$

Dividing by n^{2}-2n-24 undoes the multiplication by n^{2}-2n-24.

$$Y=\frac{n-46}{n^{2}-2n-24}$$

Divide n-46 by n^{2}-2n-24.

$$Y=\frac{n-46}{\left(n-6\right)\left(n+4\right)}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{\sqrt{100Y^{2}-180Y+1}+2Y+1}{2Y}\text{; }n=\frac{-\sqrt{100Y^{2}-180Y+1}+2Y+1}{2Y}\text{, }&Y\neq 0\\n=46\text{, }&Y=0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\sqrt{100Y^{2}-180Y+1}+2Y+1}{2Y}\text{; }n=\frac{-\sqrt{100Y^{2}-180Y+1}+2Y+1}{2Y}\text{, }&\left(Y\neq 0\text{ and }Y\leq -\frac{2\sqrt{5}}{5}+\frac{9}{10}\right)\text{ or }Y\geq \frac{2\sqrt{5}}{5}+\frac{9}{10}\\n=46\text{, }&Y=0\end{matrix}\right.$$