The sum of `(6a + 4b - c + 3), (2b - 3c + 4), (11b - 7a + 2c - 1)` and `(2c - 5a - 6)` is

Options

A. `(4a - 6b + 2)`
B. (-3a + 14b - 3c + 2)
C. `(-6a + 17b)`
D. `(-6a + 6b + c - 4)`

Answer

To find the sum of the algebraic expressions \( (6a + 4b - c + 3), (2b - 3c + 4), (11b - 7a + 2c - 1) \), and \( (2c - 5a - 6) \), we will follow these steps:

### Step 1: Write down the expressions
We have the following expressions:
1. \( 6a + 4b - c + 3 \)
2. \( 2b - 3c + 4 \)
3. \( 11b - 7a + 2c - 1 \)
4. \( 2c - 5a - 6 \)

### Step 2: Combine the first two expressions
Let's add the first and second expressions:
\[ (6a + 4b - c + 3) + (2b - 3c + 4) \]
Distributing the addition:
\[ 6a + 4b - c + 3 + 2b - 3c + 4 \]
Now, combine like terms:
- For \( a \): \( 6a \)
- For \( b \): \( 4b + 2b = 6b \)
- For \( c \): \( -c - 3c = -4c \)
- For constants: \( 3 + 4 = 7 \)

So, the result of the first two expressions is:
\[ 6a + 6b - 4c + 7 \]

### Step 3: Combine the third and fourth expressions
Now, let's add the third and fourth expressions:
\[ (11b - 7a + 2c - 1) + (2c - 5a - 6) \]
Distributing the addition:
\[ 11b - 7a + 2c - 1 + 2c - 5a - 6 \]
Now, combine like terms:
- For \( a \): \( -7a - 5a = -12a \)
- For \( b \): \( 11b \)
- For \( c \): \( 2c + 2c = 4c \)
- For constants: \( -1 - 6 = -7 \)

So, the result of the third and fourth expressions is:
\[ 11b - 12a + 4c - 7 \]

### Step 4: Combine the results from Step 2 and Step 3
Now, we need to add the results from Step 2 and Step 3:
\[ (6a + 6b - 4c + 7) + (11b - 12a + 4c - 7) \]
Distributing the addition:
\[ 6a + 6b - 4c + 7 + 11b - 12a + 4c - 7 \]
Now, combine like terms:
- For \( a \): \( 6a - 12a = -6a \)
- For \( b \): \( 6b + 11b = 17b \)
- For \( c \): \( -4c + 4c = 0 \) (these terms cancel out)
- For constants: \( 7 - 7 = 0 \) (these terms also cancel out)

### Final Result
Thus, the final result of the sum is:
\[ -6a + 17b \]

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