Solve n_{e}*30+(300-30)*frac{n_{e}}{varepsilon+1}=50

$$n _ { e } \cdot 30 + ( 300 - 30 ) \cdot \frac { n _ { e } } { \varepsilon + 1 } = 50$$

Solve for n_e

$$n_{e}=\frac{5\left(\epsilon +1\right)}{3\left(\epsilon +10\right)}$$

$$\epsilon \neq -10\text{ and }\epsilon \neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \epsilon +1.

$$n_{e}\times 30\left(\epsilon +1\right)+\left(300-30\right)n_{e}=50\left(\epsilon +1\right)$$

Use the distributive property to multiply n_{e}\times 30 by \epsilon +1.

$$30n_{e}\epsilon +n_{e}\times 30+\left(300-30\right)n_{e}=50\left(\epsilon +1\right)$$

Subtract 30 from 300 to get 270.

$$30n_{e}\epsilon +n_{e}\times 30+270n_{e}=50\left(\epsilon +1\right)$$

Combine n_{e}\times 30 and 270n_{e} to get 300n_{e}.

$$30n_{e}\epsilon +300n_{e}=50\left(\epsilon +1\right)$$

Use the distributive property to multiply 50 by \epsilon +1.

$$30n_{e}\epsilon +300n_{e}=50\epsilon +50$$

Combine all terms containing n_{e}.

$$\left(30\epsilon +300\right)n_{e}=50\epsilon +50$$

Divide both sides by 30\epsilon +300.

$$\frac{\left(30\epsilon +300\right)n_{e}}{30\epsilon +300}=\frac{50\epsilon +50}{30\epsilon +300}$$

Dividing by 30\epsilon +300 undoes the multiplication by 30\epsilon +300.

$$n_{e}=\frac{50\epsilon +50}{30\epsilon +300}$$

Divide 50+50\epsilon by 30\epsilon +300.

$$n_{e}=\frac{5\left(\epsilon +1\right)}{3\left(\epsilon +10\right)}$$

Solve for ε

$$\epsilon =-\frac{5\left(6n_{e}-1\right)}{3n_{e}-5}$$

$$n_{e}\neq 0\text{ and }n_{e}\neq \frac{5}{3}$$

Steps for Solving Linear Equation

Variable \epsilon cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \epsilon +1.

$$n_{e}\times 30\left(\epsilon +1\right)+\left(300-30\right)n_{e}=50\left(\epsilon +1\right)$$

Use the distributive property to multiply n_{e}\times 30 by \epsilon +1.

$$30n_{e}\epsilon +n_{e}\times 30+\left(300-30\right)n_{e}=50\left(\epsilon +1\right)$$

Subtract 30 from 300 to get 270.

$$30n_{e}\epsilon +n_{e}\times 30+270n_{e}=50\left(\epsilon +1\right)$$

Combine n_{e}\times 30 and 270n_{e} to get 300n_{e}.

$$30n_{e}\epsilon +300n_{e}=50\left(\epsilon +1\right)$$

Use the distributive property to multiply 50 by \epsilon +1.

$$30n_{e}\epsilon +300n_{e}=50\epsilon +50$$

Subtract 50\epsilon from both sides.

$$30n_{e}\epsilon +300n_{e}-50\epsilon =50$$

Subtract 300n_{e} from both sides.

$$30n_{e}\epsilon -50\epsilon =50-300n_{e}$$

Combine all terms containing \epsilon .

$$\left(30n_{e}-50\right)\epsilon =50-300n_{e}$$

Divide both sides by 30n_{e}-50.

$$\frac{\left(30n_{e}-50\right)\epsilon }{30n_{e}-50}=\frac{50-300n_{e}}{30n_{e}-50}$$

Dividing by 30n_{e}-50 undoes the multiplication by 30n_{e}-50.

$$\epsilon =\frac{50-300n_{e}}{30n_{e}-50}$$

Divide 50-300n_{e} by 30n_{e}-50.

$$\epsilon =\frac{5\left(1-6n_{e}\right)}{3n_{e}-5}$$

Variable \epsilon cannot be equal to -1.

$$\epsilon =\frac{5\left(1-6n_{e}\right)}{3n_{e}-5}\text{, }\epsilon \neq -1$$