To solve the equation \((x + y, 1) = (3, x - y)\), we need to compare the corresponding components of the ordered pairs.

Given:
\[
(x + y, 1) = (3, x - y)
\]

This implies two separate equations:
1. \(x + y = 3\)
2. \(1 = x - y\)

Now, let's solve these equations step by step.

### Step 1: Solve for \(x\) and \(y\)
From the second equation:
\[
1 = x - y \quad \text{(Equation 2)}
\]
We can rearrange this to express \(x\) in terms of \(y\):
\[
x = y + 1
\]

### Step 2: Substitute into the first equation
Now, substitute \(x = y + 1\) into the first equation:
\[
(y + 1) + y = 3
\]
Simplify:
\[
2y + 1 = 3
\]
Subtract 1 from both sides:
\[
2y = 2
\]
Divide by 2:
\[
y = 1
\]

### Step 3: Find \(x\)
Using \(y = 1\) in the expression for \(x\):
\[
x = 1 + 1 = 2
\]

### Final Answer
The solution is:
\[
x = 2, \quad y = 1
\]

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How do you solve systems of linear equations using substitution?
2. What happens if the two ordered pairs had different dimensions?
3. How can you solve the system if both equations were non-linear?
4. What methods can be used to solve systems of equations graphically?
5. How can you check if a solution to a system of equations is unique or if there are infinite solutions?

**Tip:** When solving systems of equations, always verify your solution by substituting it back into the original equations.

Learn how to solve the system of linear equations (x + y, 1) = (3, x - y) step-by-step. This problem involves solving for variables in a system of equations and understanding linear algebra concepts. Suitable for students in Grades 9-12.

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