Solve for a, b and c, `a + b = 10, b + c = 12 and a + c = 16`.

Answer

To solve the equations \( a + b = 10 \), \( b + c = 12 \), and \( a + c = 16 \), we can follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( a + b = 10 \) (Equation 1)
2. \( b + c = 12 \) (Equation 2)
3. \( a + c = 16 \) (Equation 3)

### Step 2: Add all three equations
Now, we add all three equations together:
\[ (a + b) + (b + c) + (a + c) = 10 + 12 + 16 \]
This simplifies to:
\[ 2a + 2b + 2c = 38 \]

### Step 3: Simplify the equation
Dividing the entire equation by 2 gives us:
\[ a + b + c = 19 \quad (Equation 4) \]

### Step 4: Solve for \( c \)
From Equation 1, we know \( a + b = 10 \). We can substitute this into Equation 4:
\[ 10 + c = 19 \]
Solving for \( c \):
\[ c = 19 - 10 = 9 \]

### Step 5: Solve for \( a \)
Now, we can use Equation 3 to find \( a \):
\[ a + c = 16 \]
Substituting \( c = 9 \):
\[ a + 9 = 16 \]
Solving for \( a \):
\[ a = 16 - 9 = 7 \]

### Step 6: Solve for \( b \)
Finally, we can use Equation 2 to find \( b \):
\[ b + c = 12 \]
Substituting \( c = 9 \):
\[ b + 9 = 12 \]
Solving for \( b \):
\[ b = 12 - 9 = 3 \]

### Final Values
Thus, the values are:
- \( a = 7 \)
- \( b = 3 \)
- \( c = 9 \)

### Summary
The solution gives us:
\[ a = 7, \quad b = 3, \quad c = 9 \]