If `a + 2b + 2c = 11, 2a + b+ 2c = 10 and 2a+ 2b+ c = 9` , What is the value of a ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( a \) from the given equations:

1. **Write down the equations:**
\[ \begin{align*} (1) & \quad a + 2b + 2c = 11 \\ (2) & \quad 2a + b + 2c = 10 \\ (3) & \quad 2a + 2b + c = 9 \end{align*} \]

2. **Subtract equation (2) from equation (1) to eliminate \( c \):**
\[ (1) - (2): \quad (a + 2b + 2c) - (2a + b + 2c) = 11 - 10 \]
This simplifies to:
\[ -a + b = 1 \quad \text{(Equation 4)} \]

3. **Now, subtract equation (3) from equation (2) to eliminate \( c \) again:**
\[ (2) - (3): \quad (2a + b + 2c) - (2a + 2b + c) = 10 - 9 \]
This simplifies to:
\[ -b + c = 1 \quad \text{(Equation 5)} \]

4. **Now we have two new equations:**
\[ \begin{align*} (4) & \quad -a + b = 1 \\ (5) & \quad -b + c = 1 \end{align*} \]

5. **From equation (4), express \( b \) in terms of \( a \):**
\[ b = a + 1 \]

6. **Substitute \( b \) into equation (5) to find \( c \):**
\[ - (a + 1) + c = 1 \]
This simplifies to:
\[ c = a + 2 \]

7. **Substitute \( b \) and \( c \) back into any of the original equations. Let's use equation (1):**
\[ a + 2(a + 1) + 2(a + 2) = 11 \]
Simplifying this:
\[ a + 2a + 2 + 2a + 4 = 11 \]
Combine like terms:
\[ 5a + 6 = 11 \]
Now, isolate \( a \):
\[ 5a = 11 - 6 \\ 5a = 5 \\ a = 1 \]

8. **Thus, the value of \( a \) is:**
\[ \boxed{1} \]