Solve y_{1}(n)-y_{1}(n-1)+1/4y_{1}(n-2)=x_{1}(n)-2x_{1}(n-1)

$$y _ { 1 } ( n ) - y _ { 1 } ( n - 1 ) + \frac { 1 } { 4 } y _ { 1 } ( n - 2 ) = x _ { 1 } ( n ) - 2 x _ { 1 } ( n - 1 )$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{2\left(y_{1}-4x_{1}\right)}{4x_{1}+y_{1}}\text{, }&y_{1}\neq -4x_{1}\\n\in \mathrm{C}\text{, }&y_{1}=0\text{ and }x_{1}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y_{1} by n-1.

$$y_{1}n-\left(y_{1}n-y_{1}\right)+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

To find the opposite of y_{1}n-y_{1}, find the opposite of each term.

$$y_{1}n-y_{1}n+y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1}n and -y_{1}n to get 0.

$$y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Use the distributive property to multiply \frac{1}{4}y_{1} by n-2.

$$y_{1}+\frac{1}{4}y_{1}n-\frac{1}{2}y_{1}=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1} and -\frac{1}{2}y_{1} to get \frac{1}{2}y_{1}.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n=x_{1}n-2x_{1}\left(n-1\right)$$

Add 2x_{1}\left(n-1\right) to both sides.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}\left(n-1\right)=x_{1}n$$

Use the distributive property to multiply 2x_{1} by n-1.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}n-2x_{1}=x_{1}n$$

Subtract x_{1}n from both sides.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}n-2x_{1}-x_{1}n=0$$

Combine 2x_{1}n and -x_{1}n to get x_{1}n.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+x_{1}n-2x_{1}=0$$

Subtract \frac{1}{2}y_{1} from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{4}y_{1}n+x_{1}n-2x_{1}=-\frac{1}{2}y_{1}$$

Add 2x_{1} to both sides.

$$\frac{1}{4}y_{1}n+x_{1}n=-\frac{1}{2}y_{1}+2x_{1}$$

Combine all terms containing n.

$$\left(\frac{1}{4}y_{1}+x_{1}\right)n=-\frac{1}{2}y_{1}+2x_{1}$$

The equation is in standard form.

$$\left(\frac{y_{1}}{4}+x_{1}\right)n=-\frac{y_{1}}{2}+2x_{1}$$

Divide both sides by x_{1}+\frac{1}{4}y_{1}.

$$\frac{\left(\frac{y_{1}}{4}+x_{1}\right)n}{\frac{y_{1}}{4}+x_{1}}=\frac{-\frac{y_{1}}{2}+2x_{1}}{\frac{y_{1}}{4}+x_{1}}$$

Dividing by x_{1}+\frac{1}{4}y_{1} undoes the multiplication by x_{1}+\frac{1}{4}y_{1}.

$$n=\frac{-\frac{y_{1}}{2}+2x_{1}}{\frac{y_{1}}{4}+x_{1}}$$

Divide -\frac{y_{1}}{2}+2x_{1} by x_{1}+\frac{1}{4}y_{1}.

$$n=\frac{2\left(4x_{1}-y_{1}\right)}{4x_{1}+y_{1}}$$

Solve for x_1 (complex solution)

$$\left\{\begin{matrix}x_{1}=-\frac{y_{1}\left(n+2\right)}{4\left(n-2\right)}\text{, }&n\neq 2\\x_{1}\in \mathrm{C}\text{, }&y_{1}=0\text{ and }n=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y_{1} by n-1.

$$y_{1}n-\left(y_{1}n-y_{1}\right)+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

To find the opposite of y_{1}n-y_{1}, find the opposite of each term.

$$y_{1}n-y_{1}n+y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1}n and -y_{1}n to get 0.

$$y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Use the distributive property to multiply \frac{1}{4}y_{1} by n-2.

$$y_{1}+\frac{1}{4}y_{1}n-\frac{1}{2}y_{1}=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1} and -\frac{1}{2}y_{1} to get \frac{1}{2}y_{1}.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n=x_{1}n-2x_{1}\left(n-1\right)$$

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

$$x_{1}n-2x_{1}\left(n-1\right)=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Use the distributive property to multiply -2x_{1} by n-1.

$$x_{1}n-2x_{1}n+2x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Combine x_{1}n and -2x_{1}n to get -x_{1}n.

$$-x_{1}n+2x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Combine all terms containing x_{1}.

$$\left(-n+2\right)x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

The equation is in standard form.

$$\left(2-n\right)x_{1}=\frac{ny_{1}}{4}+\frac{y_{1}}{2}$$

Divide both sides by -n+2.

$$\frac{\left(2-n\right)x_{1}}{2-n}=\frac{y_{1}\left(n+2\right)}{4\left(2-n\right)}$$

Dividing by -n+2 undoes the multiplication by -n+2.

$$x_{1}=\frac{y_{1}\left(n+2\right)}{4\left(2-n\right)}$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{2\left(y_{1}-4x_{1}\right)}{4x_{1}+y_{1}}\text{, }&y_{1}\neq -4x_{1}\\n\in \mathrm{R}\text{, }&y_{1}=0\text{ and }x_{1}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y_{1} by n-1.

$$y_{1}n-\left(y_{1}n-y_{1}\right)+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

To find the opposite of y_{1}n-y_{1}, find the opposite of each term.

$$y_{1}n-y_{1}n+y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1}n and -y_{1}n to get 0.

$$y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Use the distributive property to multiply \frac{1}{4}y_{1} by n-2.

$$y_{1}+\frac{1}{4}y_{1}n-\frac{1}{2}y_{1}=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1} and -\frac{1}{2}y_{1} to get \frac{1}{2}y_{1}.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n=x_{1}n-2x_{1}\left(n-1\right)$$

Add 2x_{1}\left(n-1\right) to both sides.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}\left(n-1\right)=x_{1}n$$

Use the distributive property to multiply 2x_{1} by n-1.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}n-2x_{1}=x_{1}n$$

Subtract x_{1}n from both sides.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+2x_{1}n-2x_{1}-x_{1}n=0$$

Combine 2x_{1}n and -x_{1}n to get x_{1}n.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n+x_{1}n-2x_{1}=0$$

Subtract \frac{1}{2}y_{1} from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{4}y_{1}n+x_{1}n-2x_{1}=-\frac{1}{2}y_{1}$$

Add 2x_{1} to both sides.

$$\frac{1}{4}y_{1}n+x_{1}n=-\frac{1}{2}y_{1}+2x_{1}$$

Combine all terms containing n.

$$\left(\frac{1}{4}y_{1}+x_{1}\right)n=-\frac{1}{2}y_{1}+2x_{1}$$

The equation is in standard form.

$$\left(\frac{y_{1}}{4}+x_{1}\right)n=-\frac{y_{1}}{2}+2x_{1}$$

Divide both sides by x_{1}+\frac{1}{4}y_{1}.

$$\frac{\left(\frac{y_{1}}{4}+x_{1}\right)n}{\frac{y_{1}}{4}+x_{1}}=\frac{-\frac{y_{1}}{2}+2x_{1}}{\frac{y_{1}}{4}+x_{1}}$$

Dividing by x_{1}+\frac{1}{4}y_{1} undoes the multiplication by x_{1}+\frac{1}{4}y_{1}.

$$n=\frac{-\frac{y_{1}}{2}+2x_{1}}{\frac{y_{1}}{4}+x_{1}}$$

Divide -\frac{y_{1}}{2}+2x_{1} by x_{1}+\frac{1}{4}y_{1}.

$$n=\frac{2\left(4x_{1}-y_{1}\right)}{4x_{1}+y_{1}}$$

Solve for x_1

$$\left\{\begin{matrix}x_{1}=-\frac{y_{1}\left(n+2\right)}{4\left(n-2\right)}\text{, }&n\neq 2\\x_{1}\in \mathrm{R}\text{, }&y_{1}=0\text{ and }n=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y_{1} by n-1.

$$y_{1}n-\left(y_{1}n-y_{1}\right)+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

To find the opposite of y_{1}n-y_{1}, find the opposite of each term.

$$y_{1}n-y_{1}n+y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1}n and -y_{1}n to get 0.

$$y_{1}+\frac{1}{4}y_{1}\left(n-2\right)=x_{1}n-2x_{1}\left(n-1\right)$$

Use the distributive property to multiply \frac{1}{4}y_{1} by n-2.

$$y_{1}+\frac{1}{4}y_{1}n-\frac{1}{2}y_{1}=x_{1}n-2x_{1}\left(n-1\right)$$

Combine y_{1} and -\frac{1}{2}y_{1} to get \frac{1}{2}y_{1}.

$$\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n=x_{1}n-2x_{1}\left(n-1\right)$$

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

$$x_{1}n-2x_{1}\left(n-1\right)=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Use the distributive property to multiply -2x_{1} by n-1.

$$x_{1}n-2x_{1}n+2x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Combine x_{1}n and -2x_{1}n to get -x_{1}n.

$$-x_{1}n+2x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

Combine all terms containing x_{1}.

$$\left(-n+2\right)x_{1}=\frac{1}{2}y_{1}+\frac{1}{4}y_{1}n$$

The equation is in standard form.

$$\left(2-n\right)x_{1}=\frac{ny_{1}}{4}+\frac{y_{1}}{2}$$

Divide both sides by -n+2.

$$\frac{\left(2-n\right)x_{1}}{2-n}=\frac{y_{1}\left(n+2\right)}{4\left(2-n\right)}$$

Dividing by -n+2 undoes the multiplication by -n+2.

$$x_{1}=\frac{y_{1}\left(n+2\right)}{4\left(2-n\right)}$$