Solve (n)=1/2y(n-1)+x(n)+1/2x(n-1)

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

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{x+y}{2-y-3x}\text{, }&y\neq 2-3x\\n\in \mathrm{C}\text{, }&y=-1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}x\left(n-1\right)$$

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}xn-\frac{1}{2}x$$

Combine xn and \frac{1}{2}xn to get \frac{3}{2}xn.

$$n=\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

Subtract \frac{1}{2}yn from both sides.

$$n-\frac{1}{2}yn=-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

Subtract \frac{3}{2}xn from both sides.

$$n-\frac{1}{2}yn-\frac{3}{2}xn=-\frac{1}{2}y-\frac{1}{2}x$$

Combine all terms containing n.

$$\left(1-\frac{1}{2}y-\frac{3}{2}x\right)n=-\frac{1}{2}y-\frac{1}{2}x$$

The equation is in standard form.

$$\left(-\frac{y}{2}-\frac{3x}{2}+1\right)n=\frac{-x-y}{2}$$

Divide both sides by -\frac{3}{2}x-\frac{1}{2}y+1.

$$\frac{\left(-\frac{y}{2}-\frac{3x}{2}+1\right)n}{-\frac{y}{2}-\frac{3x}{2}+1}=\frac{-x-y}{2\left(-\frac{y}{2}-\frac{3x}{2}+1\right)}$$

Dividing by -\frac{3}{2}x-\frac{1}{2}y+1 undoes the multiplication by -\frac{3}{2}x-\frac{1}{2}y+1.

$$n=\frac{-x-y}{2\left(-\frac{y}{2}-\frac{3x}{2}+1\right)}$$

Divide \frac{-y-x}{2} by -\frac{3}{2}x-\frac{1}{2}y+1.

$$n=-\frac{x+y}{2-y-3x}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{ny-y-2n}{3n-1}\text{, }&n\neq \frac{1}{3}\\x\in \mathrm{C}\text{, }&y=-1\text{ and }n=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}x\left(n-1\right)$$

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}xn-\frac{1}{2}x$$

Combine xn and \frac{1}{2}xn to get \frac{3}{2}xn.

$$n=\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

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

$$\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x=n$$

Subtract \frac{1}{2}yn from both sides.

$$-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x=n-\frac{1}{2}yn$$

Add \frac{1}{2}y to both sides.

$$\frac{3}{2}xn-\frac{1}{2}x=n-\frac{1}{2}yn+\frac{1}{2}y$$

Combine all terms containing x.

$$\left(\frac{3}{2}n-\frac{1}{2}\right)x=n-\frac{1}{2}yn+\frac{1}{2}y$$

The equation is in standard form.

$$\frac{3n-1}{2}x=-\frac{ny}{2}+\frac{y}{2}+n$$

Divide both sides by \frac{3}{2}n-\frac{1}{2}.

$$\frac{2\times \frac{3n-1}{2}x}{3n-1}=\frac{2\left(-\frac{ny}{2}+\frac{y}{2}+n\right)}{3n-1}$$

Dividing by \frac{3}{2}n-\frac{1}{2} undoes the multiplication by \frac{3}{2}n-\frac{1}{2}.

$$x=\frac{2\left(-\frac{ny}{2}+\frac{y}{2}+n\right)}{3n-1}$$

Divide n-\frac{yn}{2}+\frac{y}{2} by \frac{3}{2}n-\frac{1}{2}.

$$x=\frac{2n+y-ny}{3n-1}$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{x+y}{2-y-3x}\text{, }&y\neq 2-3x\\n\in \mathrm{R}\text{, }&y=-1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}x\left(n-1\right)$$

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}xn-\frac{1}{2}x$$

Combine xn and \frac{1}{2}xn to get \frac{3}{2}xn.

$$n=\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

Subtract \frac{1}{2}yn from both sides.

$$n-\frac{1}{2}yn=-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

Subtract \frac{3}{2}xn from both sides.

$$n-\frac{1}{2}yn-\frac{3}{2}xn=-\frac{1}{2}y-\frac{1}{2}x$$

Combine all terms containing n.

$$\left(1-\frac{1}{2}y-\frac{3}{2}x\right)n=-\frac{1}{2}y-\frac{1}{2}x$$

The equation is in standard form.

$$\left(-\frac{y}{2}-\frac{3x}{2}+1\right)n=\frac{-x-y}{2}$$

Divide both sides by -\frac{3}{2}x-\frac{1}{2}y+1.

$$\frac{\left(-\frac{y}{2}-\frac{3x}{2}+1\right)n}{-\frac{y}{2}-\frac{3x}{2}+1}=\frac{-x-y}{2\left(-\frac{y}{2}-\frac{3x}{2}+1\right)}$$

Dividing by -\frac{3}{2}x-\frac{1}{2}y+1 undoes the multiplication by -\frac{3}{2}x-\frac{1}{2}y+1.

$$n=\frac{-x-y}{2\left(-\frac{y}{2}-\frac{3x}{2}+1\right)}$$

Divide \frac{-y-x}{2} by -\frac{3}{2}x-\frac{1}{2}y+1.

$$n=-\frac{x+y}{2-y-3x}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{ny-y-2n}{3n-1}\text{, }&n\neq \frac{1}{3}\\x\in \mathrm{R}\text{, }&y=-1\text{ and }n=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}x\left(n-1\right)$$

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

$$n=\frac{1}{2}yn-\frac{1}{2}y+xn+\frac{1}{2}xn-\frac{1}{2}x$$

Combine xn and \frac{1}{2}xn to get \frac{3}{2}xn.

$$n=\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x$$

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

$$\frac{1}{2}yn-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x=n$$

Subtract \frac{1}{2}yn from both sides.

$$-\frac{1}{2}y+\frac{3}{2}xn-\frac{1}{2}x=n-\frac{1}{2}yn$$

Add \frac{1}{2}y to both sides.

$$\frac{3}{2}xn-\frac{1}{2}x=n-\frac{1}{2}yn+\frac{1}{2}y$$

Combine all terms containing x.

$$\left(\frac{3}{2}n-\frac{1}{2}\right)x=n-\frac{1}{2}yn+\frac{1}{2}y$$

The equation is in standard form.

$$\frac{3n-1}{2}x=-\frac{ny}{2}+\frac{y}{2}+n$$

Divide both sides by \frac{3}{2}n-\frac{1}{2}.

$$\frac{2\times \frac{3n-1}{2}x}{3n-1}=\frac{2\left(-\frac{ny}{2}+\frac{y}{2}+n\right)}{3n-1}$$

Dividing by \frac{3}{2}n-\frac{1}{2} undoes the multiplication by \frac{3}{2}n-\frac{1}{2}.

$$x=\frac{2\left(-\frac{ny}{2}+\frac{y}{2}+n\right)}{3n-1}$$

Divide n-\frac{yn}{2}+\frac{y}{2} by \frac{3}{2}n-\frac{1}{2}.

$$x=\frac{2n+y-ny}{3n-1}$$