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.