For what value of k, the system of equations 4x + 6y = 11 and 2x + ky = 7 will be inconsistent?

Options

A. k = 3
B. k = 4
C. no value of k is possible
D. k can have any real value

Answer

To determine the value of \( k \) for which the system of equations

1. \( 4x + 6y = 11 \)
2. \( 2x + ky = 7 \)

is inconsistent, we will follow these steps:

### Step 1: Rewrite the equations in standard form
We can rewrite both equations in the standard form \( Ax + By + C = 0 \).

- For the first equation:
\[ 4x + 6y - 11 = 0 \]
Here, \( A_1 = 4 \), \( B_1 = 6 \), and \( C_1 = -11 \).

- For the second equation:
\[ 2x + ky - 7 = 0 \]
Here, \( A_2 = 2 \), \( B_2 = k \), and \( C_2 = -7 \).

### Step 2: Use the condition for inconsistency
The system of equations is inconsistent if the following condition holds:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2} \]

Substituting the values we have:
\[ \frac{4}{2} = \frac{6}{k} \neq \frac{-11}{-7} \]

### Step 3: Simplify the ratios
Calculating the left side:
\[ \frac{4}{2} = 2 \]

Calculating the right side:
\[ \frac{-11}{-7} = \frac{11}{7} \]

Now we have:
\[ 2 = \frac{6}{k} \quad \text{and} \quad 2 \neq \frac{11}{7} \]

### Step 4: Solve for \( k \)
From the equation \( 2 = \frac{6}{k} \), we can cross-multiply to find \( k \):
\[ 2k = 6 \implies k = \frac{6}{2} = 3 \]

### Step 5: Verify the second condition
Now we need to check if \( 2 \neq \frac{11}{7} \):
\[ 2 = \frac{14}{7} \quad \text{and} \quad \frac{14}{7} \neq \frac{11}{7} \]
This condition holds true.

### Conclusion
Thus, the value of \( k \) for which the system of equations is inconsistent is:
\[ \boxed{3} \]