If ` a + 2b + 3c = 24` and `3a + 2b + c = 36` what is the value of `(a+b+c)` ?

Options

A. 8
B. 12
C. 15
D. 60

Answer

To solve the equations \( a + 2b + 3c = 24 \) and \( 3a + 2b + c = 36 \) and find the value of \( a + b + c \), we can follow these steps:

### Step 1: Write down the equations
We have the two equations:
1. \( a + 2b + 3c = 24 \) (Equation 1)
2. \( 3a + 2b + c = 36 \) (Equation 2)

### Step 2: Subtract Equation 1 from Equation 2
We will subtract Equation 1 from Equation 2 to eliminate \( b \):
\[ (3a + 2b + c) - (a + 2b + 3c) = 36 - 24 \]
This simplifies to:
\[ 3a - a + 2b - 2b + c - 3c = 12 \]
\[ 2a - 2c = 12 \]
Dividing the entire equation by 2 gives:
\[ a - c = 6 \quad \text{(Equation 3)} \]

### Step 3: Express \( a \) in terms of \( c \)
From Equation 3, we can express \( a \) as:
\[ a = 6 + c \quad \text{(Equation 4)} \]

### Step 4: Substitute \( a \) in Equation 1
Now we will substitute Equation 4 into Equation 1:
\[ (6 + c) + 2b + 3c = 24 \]
This simplifies to:
\[ 6 + c + 2b + 3c = 24 \]
Combining like terms gives:
\[ 6 + 4c + 2b = 24 \]
Subtracting 6 from both sides:
\[ 4c + 2b = 18 \]
Dividing the entire equation by 2 gives:
\[ 2c + b = 9 \quad \text{(Equation 5)} \]

### Step 5: Express \( b \) in terms of \( c \)
From Equation 5, we can express \( b \) as:
\[ b = 9 - 2c \quad \text{(Equation 6)} \]

### Step 6: Find \( a + b + c \)
Now we can find \( a + b + c \) using Equations 4 and 6:
\[ a + b + c = (6 + c) + (9 - 2c) + c \]
Combining the terms:
\[ = 6 + c + 9 - 2c + c \]
This simplifies to:
\[ = 6 + 9 + 0c = 15 \]

### Final Answer
Thus, the value of \( a + b + c \) is \( \boxed{15} \).