Given the system of equations:

1. \( a + b = 1 \)
2. \( b + c = 2 \)
3. \( c + a = 3 \)

We want to find \( a + b + c \).

### Step 1: Add all three equations
Add the left-hand sides and the right-hand sides of the equations:

\[
(a + b) + (b + c) + (c + a) = 1 + 2 + 3
\]

Simplify the left-hand side:

\[
2a + 2b + 2c = 6
\]

### Step 2: Solve for \( a + b + c \)
Now, factor out the 2 on the left-hand side:

\[
2(a + b + c) = 6
\]

Divide both sides by 2:

\[
a + b + c = 3
\]

### Final Answer:
\[
a + b + c = 3
\]

Would you like more details or have any questions? Here are five related questions to consider:

1. How can we solve for \(a\), \(b\), and \(c\) individually from this system of equations?
2. How do we approach systems of three-variable equations in general?
3. Can the same method be used for non-linear systems of equations?
4. What happens if one of the equations is dependent on the others?
5. Can this method of solving systems be extended to more than three equations?

**Tip:** Always check if adding or subtracting equations in a system can simplify the process.

Solve the system of linear equations: a + b = 1, b + c = 2, and c + a = 3. Learn how to simplify the problem by adding the equations and find the value of a + b + c in this detailed step-by-step solution. Suitable for Grades 7-10.

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