We are given two equations:

1. \( y = 1 - x \)
2. \( x^2 + y^2 = 1 \)

This is a system of equations involving a line and a circle. Let's solve this system step by step.

### Step 1: Substituting \( y = 1 - x \) into the second equation
Substitute \( y = 1 - x \) into the equation \( x^2 + y^2 = 1 \):

\[
x^2 + (1 - x)^2 = 1
\]

### Step 2: Expanding the equation
Now, expand \( (1 - x)^2 \):

\[
(1 - x)^2 = 1 - 2x + x^2
\]

So the equation becomes:

\[
x^2 + (1 - 2x + x^2) = 1
\]

### Step 3: Simplifying the equation
Combine like terms:

\[
x^2 + 1 - 2x + x^2 = 1
\]

\[
2x^2 - 2x + 1 = 1
\]

### Step 4: Solving for \( x \)
Subtract 1 from both sides:

\[
2x^2 - 2x = 0
\]

Factor the equation:

\[
2x(x - 1) = 0
\]

So, \( x = 0 \) or \( x = 1 \).

### Step 5: Finding corresponding \( y \) values
For each value of \( x \), find the corresponding \( y \) using the first equation \( y = 1 - x \).

- When \( x = 0 \):
  \[
  y = 1 - 0 = 1
  \]
  
- When \( x = 1 \):
  \[
  y = 1 - 1 = 0
  \]

### Final Solutions
The solutions to the system are the points:
- \( (0, 1) \)
- \( (1, 0) \)

These points represent the intersections of the line \( y = 1 - x \) and the circle \( x^2 + y^2 = 1 \).

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Would you like more details on any step or have any questions?

Here are some related questions to explore:
1. What is the geometric interpretation of the system of equations?
2. Can we find the area enclosed between the line and the circle?
3. How can you solve the system using substitution with another method?
4. What happens if you change the radius of the circle?
5. What if the equation of the line changes to \( y = 2 - x \)?

**Tip**: Always double-check factored solutions by substituting them back into the original equations.

Explore how to solve the system of equations involving the line y=1-x and the circle x² + y²=1. This problem focuses on substituting linear equations into quadratic forms, simplifying and solving for x and y. Learn the step-by-step solution, suitable for students in Grades 9-11.

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