Confused with a false proof I made 
Let $x$ and $y$ be numbers satisfying
  \begin{cases}
x+2y=4 \\
2x + y = 6 \\
x= y +1
\end{cases}
  Find $x$ and  $y$.

$x + 2y = 4$ is equivalent to $x=4-2y$
We substitute $x$ in the second equation
$$
2x+y=6 \\
8-4y+y=6 \\
-3y=-2 \\
y=2/3
$$
But $x =4-2y=4-4/3=8/3$ and we have $x = y +2=2/3+1=5/3$
which gives $y=2/3=5/3$.
WHAT?!
I'm in grade 9 and studying equations with two unknowns and thought about this example in my head and that's confusing me.

What you did proves that the system has no solution. It's just that.
