To solve the given system of equations, we have:

1. \( x - y = 1 - a \)  \(\quad\quad\quad(1)\)
2. \( x + y = 1 + a \)  \(\quad\quad\quad(2)\)

### Step 1: Add the Equations

Adding Equation (1) and Equation (2):

\[ (x - y) + (x + y) = (1 - a) + (1 + a) \]
\[ x - y + x + y = 1 - a + 1 + a \]
\[ 2x = 2 \]
\[ x = 1 \]

### Step 2: Substitute \( x = 1 \) into One of the Equations

Substitute \( x = 1 \) into Equation (1):

\[ 1 - y = 1 - a \]
\[ -y = -a \]
\[ y = a \]

### Final Solution
The solution to the system is:
\[ x = 1 \]
\[ y = a \]

Would you like more details on any of the steps or have any questions?

### Related Questions
1. How can we check the solution by substituting back into both original equations?
2. What is the geometric interpretation of these equations?
3. How would the solution change if \( a \) were a different constant?
4. Can you solve the system using matrix methods?
5. What happens if \( a \) is zero?
6. How do the solutions change if \( a \) is negative?
7. Can we find the determinant of the coefficient matrix of this system?
8. How does this system compare to other linear systems with similar structures?

### Tip
When solving systems of linear equations, adding or subtracting the equations to eliminate one variable is a very effective strategy.

Learn how to solve the system of equations x - y = 1 - a and x + y = 1 + a with a step-by-step solution. This problem covers systems of linear equations and methods such as substitution and addition. Suitable for students in Grades 9-12.

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