1.2 Solve for $$ x $$ and $$ y $$ simultaneously: $$ x+2=2 y $$ and $$ \frac{1}{x}+\frac{1}{y}=1 $$

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Solve the system of equations $$ x+2=2y; \frac{1}{x}+\frac{1}{y}=1 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+2=2y\\frac{1}{x}+\frac{1}{y}=1\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=2y-2\\frac{1}{x}+\frac{1}{y}=1\end{array}ight.$$
- step2: Substitute the value of $$x:$$
  $$\frac{1}{2y-2}+\frac{1}{y}=1$$
- step3: Multiply both sides of the equation by LCD:
  $$\left(\frac{1}{2y-2}+\frac{1}{y}ight)	imes 2y\left(y-1ight)=1	imes 2y\left(y-1ight)$$
- step4: Simplify the equation:
  $$3y-2=2y^{2}-2y$$
- step5: Move the expression to the left side:
  $$3y-2-\left(2y^{2}-2yight)=0$$
- step6: Subtract the terms:
  $$5y-2-2y^{2}=0$$
- step7: Factor the expression:
  $$\left(-2+yight)\left(1-2yight)=0$$
- step8: Separate into possible cases:
  $$\begin{align}&-2+y=0\&1-2y=0\end{align}$$
- step9: Solve the equation:
  $$\begin{align}&y=2\&y=\frac{1}{2}\end{align}$$
- step10: Calculate:
  $$y=2\cup y=\frac{1}{2}$$
- step11: Rearrange the terms:
  $$\left\{ \begin{array}{l}x=2y-2\y=2\end{array}ight.\cup \left\{ \begin{array}{l}x=2y-2\y=\frac{1}{2}\end{array}ight.$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=2\y=2\end{array}ight.\cup \left\{ \begin{array}{l}x=-1\y=\frac{1}{2}\end{array}ight.$$
- step13: Calculate:
  $$\left\{ \begin{array}{l}x=-1\y=\frac{1}{2}\end{array}ight.\cup \left\{ \begin{array}{l}x=2\y=2\end{array}ight.$$
- step14: Check the solution:
  $$\left\{ \begin{array}{l}x=-1\y=\frac{1}{2}\end{array}ight.\cup \left\{ \begin{array}{l}x=2\y=2\end{array}ight.$$
- step15: Rewrite:
  $$\left(x,yight) = \left(-1,\frac{1}{2}ight)\cup \left(x,yight) = \left(2,2ight)$$
The solutions to the system of equations $$ x+2=2y $$ and $$ \frac{1}{x}+\frac{1}{y}=1 $$ are:

1. $$ x = -1 $$ and $$ y = \frac{1}{2} $$
2. $$ x = 2 $$ and $$ y = 2 $$
The solutions are $$ x = -1, y = \frac{1}{2} $$ and $$ x = 2, y = 2 $$.

1.2 Solve for and simultaneously:
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