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!

Explore how to solve the linear equation 1 - 2x = 1 + y by isolating variables and simplifying the equation. This problem covers fundamental algebraic concepts, including solving for y in terms of x. Suitable for students in Grades 6-8.

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