Solve C(θ-θ_{1})+m_{1}C(θ-θ_{1})+m_{1}C_{x}(θ-θ_{1})+m_{N}C(θ-θ_{1})=0

$$C ( \theta - \theta _ { 1 } ) + m _ { 1 } C ( \theta - \theta _ { 1 } ) + m _ { 1 } C _ { x } ( \theta - \theta _ { 1 } ) + m _ { N } C ( \theta - \theta _ { 1 } ) = 0$$

Solve for C (complex solution)

$$\left\{\begin{matrix}C=-\frac{C_{x}m_{1}}{m_{1}+m_{N}+1}\text{, }&m_{1}\neq -\left(m_{N}+1\right)\\C\in \mathrm{C}\text{, }&\theta =\theta _{1}\text{ or }\left(m_{1}=0\text{ and }m_{N}=-1\right)\text{ or }\left(C_{x}=0\text{ and }m_{1}=-\left(m_{N}+1\right)\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\left(\theta -\theta _{1}\right)+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C_{x} by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{N}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=0$$

Subtract m_{1}C_{x}\theta from both sides. Anything subtracted from zero gives its negation.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}-m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-m_{1}C_{x}\theta$$

Add m_{1}C_{x}\theta _{1} to both sides.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-m_{1}C_{x}\theta +m_{1}C_{x}\theta _{1}$$

Reorder the terms.

$$Cm_{1}\theta -Cm_{1}\theta _{1}+Cm_{N}\theta -Cm_{N}\theta _{1}+C\theta -C\theta _{1}=C_{x}m_{1}\theta _{1}-C_{x}m_{1}\theta$$

Combine all terms containing C.

$$\left(m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}\right)C=C_{x}m_{1}\theta _{1}-C_{x}m_{1}\theta$$

Divide both sides by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$\frac{\left(m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}\right)C}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}=\frac{C_{x}m_{1}\left(\theta _{1}-\theta \right)}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}$$

Dividing by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1} undoes the multiplication by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$C=\frac{C_{x}m_{1}\left(\theta _{1}-\theta \right)}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}$$

Divide m_{1}C_{x}\left(\theta _{1}-\theta \right) by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$C=-\frac{C_{x}m_{1}}{m_{1}+m_{N}+1}$$

Solve for C_x (complex solution)

$$\left\{\begin{matrix}C_{x}=-\frac{C\left(m_{1}+m_{N}+1\right)}{m_{1}}\text{, }&m_{1}\neq 0\\C_{x}\in \mathrm{C}\text{, }&\theta =\theta _{1}\text{ or }\left(C=0\text{ and }m_{1}=0\right)\text{ or }\left(m_{1}=0\text{ and }m_{N}=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\left(\theta -\theta _{1}\right)+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C_{x} by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{N}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=0$$

Subtract C\theta from both sides. Anything subtracted from zero gives its negation.

$$-C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta$$

Add C\theta _{1} to both sides.

$$m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}$$

Subtract m_{1}C\theta from both sides.

$$-m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta$$

Add m_{1}C\theta _{1} to both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}$$

Subtract m_{N}C\theta from both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}-m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}-m_{N}C\theta$$

Add m_{N}C\theta _{1} to both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}-m_{N}C\theta +m_{N}C\theta _{1}$$

Reorder the terms.

$$C_{x}m_{1}\theta -C_{x}m_{1}\theta _{1}=Cm_{1}\theta _{1}-Cm_{1}\theta +Cm_{N}\theta _{1}-Cm_{N}\theta +C\theta _{1}-C\theta$$

Combine all terms containing C_{x}.

$$\left(m_{1}\theta -m_{1}\theta _{1}\right)C_{x}=Cm_{1}\theta _{1}-Cm_{1}\theta +Cm_{N}\theta _{1}-Cm_{N}\theta +C\theta _{1}-C\theta$$

Divide both sides by m_{1}\theta -m_{1}\theta _{1}.

$$\frac{\left(m_{1}\theta -m_{1}\theta _{1}\right)C_{x}}{m_{1}\theta -m_{1}\theta _{1}}=\frac{C\left(\theta _{1}-\theta \right)\left(m_{1}+m_{N}+1\right)}{m_{1}\theta -m_{1}\theta _{1}}$$

Dividing by m_{1}\theta -m_{1}\theta _{1} undoes the multiplication by m_{1}\theta -m_{1}\theta _{1}.

$$C_{x}=\frac{C\left(\theta _{1}-\theta \right)\left(m_{1}+m_{N}+1\right)}{m_{1}\theta -m_{1}\theta _{1}}$$

Divide C\left(1+m_{1}+m_{N}\right)\left(-\theta +\theta _{1}\right) by m_{1}\theta -m_{1}\theta _{1}.

$$C_{x}=-\frac{C\left(m_{1}+m_{N}+1\right)}{m_{1}}$$

Solve for C

$$\left\{\begin{matrix}C=-\frac{C_{x}m_{1}}{m_{1}+m_{N}+1}\text{, }&m_{1}\neq -\left(m_{N}+1\right)\\C\in \mathrm{R}\text{, }&\theta =\theta _{1}\text{ or }\left(m_{1}=0\text{ and }m_{N}=-1\right)\text{ or }\left(C_{x}=0\text{ and }m_{1}=-\left(m_{N}+1\right)\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\left(\theta -\theta _{1}\right)+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C_{x} by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{N}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=0$$

Subtract m_{1}C_{x}\theta from both sides. Anything subtracted from zero gives its negation.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}-m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-m_{1}C_{x}\theta$$

Add m_{1}C_{x}\theta _{1} to both sides.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-m_{1}C_{x}\theta +m_{1}C_{x}\theta _{1}$$

Reorder the terms.

$$Cm_{1}\theta -Cm_{1}\theta _{1}+Cm_{N}\theta -Cm_{N}\theta _{1}+C\theta -C\theta _{1}=C_{x}m_{1}\theta _{1}-C_{x}m_{1}\theta$$

Combine all terms containing C.

$$\left(m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}\right)C=C_{x}m_{1}\theta _{1}-C_{x}m_{1}\theta$$

Divide both sides by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$\frac{\left(m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}\right)C}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}=\frac{C_{x}m_{1}\left(\theta _{1}-\theta \right)}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}$$

Dividing by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1} undoes the multiplication by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$C=\frac{C_{x}m_{1}\left(\theta _{1}-\theta \right)}{m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}}$$

Divide m_{1}C_{x}\left(\theta _{1}-\theta \right) by m_{1}\theta -m_{1}\theta _{1}+m_{N}\theta -m_{N}\theta _{1}+\theta -\theta _{1}.

$$C=-\frac{C_{x}m_{1}}{m_{1}+m_{N}+1}$$

Solve for C_x

$$\left\{\begin{matrix}C_{x}=-\frac{C\left(m_{1}+m_{N}+1\right)}{m_{1}}\text{, }&m_{1}\neq 0\\C_{x}\in \mathrm{R}\text{, }&\theta =\theta _{1}\text{ or }\left(C=0\text{ and }m_{1}=0\right)\text{ or }\left(m_{1}=0\text{ and }m_{N}=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\left(\theta -\theta _{1}\right)+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\left(\theta -\theta _{1}\right)+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{1}C_{x} by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\left(\theta -\theta _{1}\right)=0$$

Use the distributive property to multiply m_{N}C by \theta -\theta _{1}.

$$C\theta -C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=0$$

Subtract C\theta from both sides. Anything subtracted from zero gives its negation.

$$-C\theta _{1}+m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta$$

Add C\theta _{1} to both sides.

$$m_{1}C\theta -m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}$$

Subtract m_{1}C\theta from both sides.

$$-m_{1}C\theta _{1}+m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta$$

Add m_{1}C\theta _{1} to both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}+m_{N}C\theta -m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}$$

Subtract m_{N}C\theta from both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}-m_{N}C\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}-m_{N}C\theta$$

Add m_{N}C\theta _{1} to both sides.

$$m_{1}C_{x}\theta -m_{1}C_{x}\theta _{1}=-C\theta +C\theta _{1}-m_{1}C\theta +m_{1}C\theta _{1}-m_{N}C\theta +m_{N}C\theta _{1}$$

Reorder the terms.

$$C_{x}m_{1}\theta -C_{x}m_{1}\theta _{1}=Cm_{1}\theta _{1}-Cm_{1}\theta +Cm_{N}\theta _{1}-Cm_{N}\theta +C\theta _{1}-C\theta$$

Combine all terms containing C_{x}.

$$\left(m_{1}\theta -m_{1}\theta _{1}\right)C_{x}=Cm_{1}\theta _{1}-Cm_{1}\theta +Cm_{N}\theta _{1}-Cm_{N}\theta +C\theta _{1}-C\theta$$

Divide both sides by m_{1}\theta -m_{1}\theta _{1}.

$$\frac{\left(m_{1}\theta -m_{1}\theta _{1}\right)C_{x}}{m_{1}\theta -m_{1}\theta _{1}}=\frac{C\left(\theta _{1}-\theta \right)\left(m_{1}+m_{N}+1\right)}{m_{1}\theta -m_{1}\theta _{1}}$$

Dividing by m_{1}\theta -m_{1}\theta _{1} undoes the multiplication by m_{1}\theta -m_{1}\theta _{1}.

$$C_{x}=\frac{C\left(\theta _{1}-\theta \right)\left(m_{1}+m_{N}+1\right)}{m_{1}\theta -m_{1}\theta _{1}}$$

Divide C\left(1+m_{1}+m_{N}\right)\left(-\theta +\theta _{1}\right) by m_{1}\theta -m_{1}\theta _{1}.

$$C_{x}=-\frac{C\left(m_{1}+m_{N}+1\right)}{m_{1}}$$