To solve the system of equations, we can follow these steps:

Given equations:
1. \( a + b = 33 \)
2. \( b + c = 25 \)
3. \( a + c = 22 \)

### Step 1: Add all three equations
Let's add all three equations together:
\[
(a + b) + (b + c) + (a + c) = 33 + 25 + 22
\]
\[
2a + 2b + 2c = 80
\]
\[
a + b + c = 40 \quad \text{(Divide by 2)}
\]

### Step 2: Solve for each variable
Now, subtract each original equation from the sum \( a + b + c = 40 \) to find the values of \( a \), \( b \), and \( c \):

1. Subtract \( a + b = 33 \) from \( a + b + c = 40 \):
\[
c = 40 - 33 = 7
\]

2. Subtract \( b + c = 25 \) from \( a + b + c = 40 \):
\[
a = 40 - 25 = 15
\]

3. Subtract \( a + c = 22 \) from \( a + b + c = 40 \):
\[
b = 40 - 22 = 18
\]

### Solution:
\[
a = 15, \quad b = 18, \quad c = 7
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would you solve a system of equations with more than three variables?
2. What methods can be used to solve non-linear systems of equations?
3. How can substitution be used to solve a system of equations?
4. What is the geometric interpretation of solving a system of linear equations?
5. How can you verify the solution to a system of equations?

**Tip:** When solving systems of equations, always check the solution by substituting the values back into the original equations.

Learn how to solve a system of linear equations: a + b = 33, b + c = 25, a + c = 22. This problem involves using algebraic methods to find the values of variables a, b, and c. Suitable for high school students studying algebra.

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