If `4a + 5b + 9c = 36` and `7a + 9b + 17c = 66`, then `a + b + c =` ________.

Answer

To solve the equations \(4a + 5b + 9c = 36\) and \(7a + 9b + 17c = 66\) and find the value of \(a + b + c\), we can follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(4a + 5b + 9c = 36\) (Equation 1)
2. \(7a + 9b + 17c = 66\) (Equation 2)

### Step 2: Subtract Equation 1 from Equation 2
We will subtract Equation 1 from Equation 2 to eliminate some variables:
\[ (7a + 9b + 17c) - (4a + 5b + 9c) = 66 - 36 \]

### Step 3: Simplify the subtraction
Now, we simplify the left-hand side:
\[ (7a - 4a) + (9b - 5b) + (17c - 9c) = 30 \]
This simplifies to:
\[ 3a + 4b + 8c = 30 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 3 from Equation 1
Next, we will subtract Equation 3 from Equation 1:
\[ (4a + 5b + 9c) - (3a + 4b + 8c) = 36 - 30 \]

### Step 5: Simplify the subtraction
Now, we simplify the left-hand side:
\[ (4a - 3a) + (5b - 4b) + (9c - 8c) = 6 \]
This simplifies to:
\[ a + b + c = 6 \]

### Final Answer
Thus, the value of \(a + b + c\) is:
\[ \boxed{6} \]