Solve varepsilon_{0}=8.854*10^-12CN^-1m^-2

$$\varepsilon _ { 0 } = 8.854 \times 10 ^ { - 12 } CN ^ { - 1 } m ^ { - 2 }$$

Solve for C

$$C=\frac{500000000000000N\epsilon _{0}m^{2}}{4427}$$

$$N\neq 0\text{ and }m\neq 0$$

Steps for Solving Linear Equation

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\epsilon _{0}=8.854\times \frac{1}{1000000000000}CN^{-1}m^{-2}$$

Multiply 8.854 and \frac{1}{1000000000000} to get \frac{4427}{500000000000000}.

$$\epsilon _{0}=\frac{4427}{500000000000000}CN^{-1}m^{-2}$$

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

$$\frac{4427}{500000000000000}CN^{-1}m^{-2}=\epsilon _{0}$$

Reorder the terms.

$$\frac{4427}{500000000000000}m^{-2}\times \frac{1}{N}C=\epsilon _{0}$$

Multiply both sides of the equation by 500000000000000N, the least common multiple of 500000000000000,N.

$$\frac{4427}{500000000000000}m^{-2}\times 500000000000000\times 1C=\epsilon _{0}\times 500000000000000N$$

Multiply \frac{4427}{500000000000000} and 500000000000000 to get 4427.

$$4427m^{-2}\times 1C=\epsilon _{0}\times 500000000000000N$$

Multiply 4427 and 1 to get 4427.

$$4427m^{-2}C=\epsilon _{0}\times 500000000000000N$$

The equation is in standard form.

$$\frac{4427}{m^{2}}C=500000000000000N\epsilon _{0}$$

Divide both sides by 4427m^{-2}.

$$\frac{\frac{4427}{m^{2}}Cm^{2}}{4427}=\frac{500000000000000N\epsilon _{0}m^{2}}{4427}$$

Dividing by 4427m^{-2} undoes the multiplication by 4427m^{-2}.

$$C=\frac{500000000000000N\epsilon _{0}m^{2}}{4427}$$

Solve for N

$$\left\{\begin{matrix}\\N\neq 0\text{, }&\text{unconditionally}\\N=\frac{4427C}{500000000000000\epsilon _{0}m^{2}}\text{, }&C\neq 0\text{ and }m\neq 0\text{ and }\epsilon _{0}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\epsilon _{0}=8.854\times \frac{1}{1000000000000}CN^{-1}m^{-2}$$

Multiply 8.854 and \frac{1}{1000000000000} to get \frac{4427}{500000000000000}.

$$\epsilon _{0}=\frac{4427}{500000000000000}CN^{-1}m^{-2}$$

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

$$\frac{4427}{500000000000000}CN^{-1}m^{-2}=\epsilon _{0}$$

Reorder the terms.

$$\frac{4427}{500000000000000}m^{-2}\times \frac{1}{N}C=\epsilon _{0}$$

Variable N cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 500000000000000N, the least common multiple of 500000000000000,N.

$$\frac{4427}{500000000000000}m^{-2}\times 500000000000000\times 1C=\epsilon _{0}\times 500000000000000N$$

Multiply \frac{4427}{500000000000000} and 500000000000000 to get 4427.

$$4427m^{-2}\times 1C=\epsilon _{0}\times 500000000000000N$$

Multiply 4427 and 1 to get 4427.

$$4427m^{-2}C=\epsilon _{0}\times 500000000000000N$$

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

$$\epsilon _{0}\times 500000000000000N=4427m^{-2}C$$

The equation is in standard form.

$$500000000000000\epsilon _{0}N=\frac{4427C}{m^{2}}$$

Divide both sides by 500000000000000\epsilon _{0}.

$$\frac{500000000000000\epsilon _{0}N}{500000000000000\epsilon _{0}}=\frac{4427C}{m^{2}\times 500000000000000\epsilon _{0}}$$

Dividing by 500000000000000\epsilon _{0} undoes the multiplication by 500000000000000\epsilon _{0}.

$$N=\frac{4427C}{m^{2}\times 500000000000000\epsilon _{0}}$$

Divide \frac{4427C}{m^{2}} by 500000000000000\epsilon _{0}.

$$N=\frac{4427C}{500000000000000\epsilon _{0}m^{2}}$$

Variable N cannot be equal to 0.

$$N=\frac{4427C}{500000000000000\epsilon _{0}m^{2}}\text{, }N\neq 0$$