Solve 2piN_{1}(1-x)=2piN_{2}(1-y)

$$2 \pi N _ { 1 } ( 1 - x ) = 2 \pi N _ { 2 } ( 1 - y )$$

Solve for N_1

$$\left\{\begin{matrix}N_{1}=-\frac{N_{2}\left(y-1\right)}{1-x}\text{, }&x\neq 1\\N_{1}\in \mathrm{R}\text{, }&\left(N_{2}=0\text{ or }y=1\right)\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Cancel out 2\pi on both sides.

$$N_{1}\left(1-x\right)=N_{2}\left(1-y\right)$$

Use the distributive property to multiply N_{1} by 1-x.

$$N_{1}-N_{1}x=N_{2}\left(1-y\right)$$

Use the distributive property to multiply N_{2} by 1-y.

$$N_{1}-N_{1}x=N_{2}-N_{2}y$$

Combine all terms containing N_{1}.

$$\left(1-x\right)N_{1}=N_{2}-N_{2}y$$

Divide both sides by -x+1.

$$\frac{\left(1-x\right)N_{1}}{1-x}=\frac{N_{2}-N_{2}y}{1-x}$$

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

$$N_{1}=\frac{N_{2}-N_{2}y}{1-x}$$

Divide N_{2}-N_{2}y by -x+1.

$$N_{1}=\frac{N_{2}\left(1-y\right)}{1-x}$$

Solve for N_2

$$\left\{\begin{matrix}N_{2}=-\frac{N_{1}\left(1-x\right)}{y-1}\text{, }&y\neq 1\\N_{2}\in \mathrm{R}\text{, }&\left(N_{1}=0\text{ or }x=1\right)\text{ and }y=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Cancel out 2\pi on both sides.

$$N_{1}\left(1-x\right)=N_{2}\left(1-y\right)$$

Use the distributive property to multiply N_{1} by 1-x.

$$N_{1}-N_{1}x=N_{2}\left(1-y\right)$$

Use the distributive property to multiply N_{2} by 1-y.

$$N_{1}-N_{1}x=N_{2}-N_{2}y$$

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

$$N_{2}-N_{2}y=N_{1}-N_{1}x$$

Combine all terms containing N_{2}.

$$\left(1-y\right)N_{2}=N_{1}-N_{1}x$$

Divide both sides by 1-y.

$$\frac{\left(1-y\right)N_{2}}{1-y}=\frac{N_{1}-N_{1}x}{1-y}$$

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

$$N_{2}=\frac{N_{1}-N_{1}x}{1-y}$$

Divide N_{1}-N_{1}x by 1-y.

$$N_{2}=\frac{N_{1}\left(1-x\right)}{1-y}$$