We are given the equation:

$1 - 2x = 1 + y$

We can solve for either $x$ or $y$, depending on which variable you want to isolate. Let's solve for $y$ first.

Steps:

1. Subtract 1 from both sides: $1 - 2x - 1 = 1 + y - 1$ This simplifies to: $-2x = y$

Thus, the solution is: $y = -2x$

If you'd like to solve for $x$, we can rearrange the equation and solve for $x$ instead. Would you like to see that?

Do you have any questions or need further details?

Here are some related questions:

1. What is the inverse function of $y = -2x$?
2. How do we solve for $x$ in terms of $y$?
3. What happens graphically when $y = -2x$?
4. Can we express this equation in slope-intercept form?
5. How can we check the solution for different values of $x$ and $y$?

Tip: Always check your final equation by plugging in values to verify its correctness!