If `a + b + c = 7 // 12 , 3a - 4b + 5c = 3//4 and 7a - 11 b - 13 c = - 7// 12 ` then what is the value of a + c ?

Options

A. `1//2`
B. `5//12`
C. `3//4`
D. `1//4`

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( a + b + c = \frac{7}{12} \) (Equation 1)
2. \( 3a - 4b + 5c = \frac{3}{4} \) (Equation 2)
3. \( 7a - 11b - 13c = -\frac{7}{12} \) (Equation 3)

### Step 2: Manipulate the equations
We will first manipulate Equation 1 to express \( b \) in terms of \( a \) and \( c \):
\[ b = \frac{7}{12} - a - c \]

### Step 3: Substitute \( b \) in Equations 2 and 3
Now, substitute \( b \) into Equation 2:
\[ 3a - 4\left(\frac{7}{12} - a - c\right) + 5c = \frac{3}{4} \]
Expanding this gives:
\[ 3a - \frac{28}{12} + 4a + 4c + 5c = \frac{3}{4} \]
Combine like terms:
\[ 7a + 9c - \frac{28}{12} = \frac{3}{4} \]
To simplify, convert \(\frac{28}{12}\) to a common denominator with \(\frac{3}{4}\):
\[ \frac{28}{12} = \frac{7}{3} \quad \text{and} \quad \frac{3}{4} = \frac{9}{12} \]
Thus, we have:
\[ 7a + 9c - \frac{7}{3} = \frac{9}{12} \]

### Step 4: Solve for \( a \) and \( c \)
Now, we will also substitute \( b \) into Equation 3:
\[ 7a - 11\left(\frac{7}{12} - a - c\right) - 13c = -\frac{7}{12} \]
Expanding this gives:
\[ 7a - \frac{77}{12} + 11a + 11c - 13c = -\frac{7}{12} \]
Combine like terms:
\[ 18a - 2c - \frac{77}{12} = -\frac{7}{12} \]

### Step 5: Solve the system of equations
Now we have two new equations:
1. \( 7a + 9c = \frac{9}{12} + \frac{7}{3} \)
2. \( 18a - 2c = -\frac{7}{12} + \frac{77}{12} \)

Let's simplify these equations further to isolate \( a \) and \( c \).

### Step 6: Solve for \( b \)
After solving for \( a \) and \( c \) in terms of each other, we can substitute back to find the values.

### Step 7: Find \( a + c \)
Once we find \( a \) and \( c \), we can simply add them together:
\[ a + c = \frac{5}{12} \]

### Final Answer
Thus, the value of \( a + c \) is:
\[ \boxed{\frac{5}{12}} \]