The pair of equations are 7x+8ky-16=0 and 14x+112y-21=0. Find the value of ‘k’ for which the system is inconsistent.

Options

A. 2
B. 3
C. 5
D. 7

Answer

To find the value of \( k \) for which the system of equations is inconsistent, we need to analyze the given equations:

1. **Equations**:
\[ 7x + 8ky - 16 = 0 \quad \text{(Equation 1)} \]
\[ 14x + 112y - 21 = 0 \quad \text{(Equation 2)} \]

2. **Identify Coefficients**:
From Equation 1:
- Coefficient of \( x \) (denoted as \( a_1 \)) = 7
- Coefficient of \( y \) (denoted as \( b_1 \)) = \( 8k \)

From Equation 2:
- Coefficient of \( x \) (denoted as \( a_2 \)) = 14
- Coefficient of \( y \) (denoted as \( b_2 \)) = 112

3. **Condition for Inconsistency**:
A system of equations is inconsistent if the ratios of the coefficients of \( x \) and \( y \) are equal, but the constant terms are not. This can be expressed as:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]
Substituting the values we have:
\[ \frac{7}{14} = \frac{8k}{112} \]

4. **Simplifying the Ratios**:
Simplifying the left side:
\[ \frac{7}{14} = \frac{1}{2} \]

Now, simplifying the right side:
\[ \frac{8k}{112} = \frac{k}{14} \]

5. **Setting Up the Equation**:
Now we set the two simplified ratios equal to each other:
\[ \frac{1}{2} = \frac{k}{14} \]

6. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 1 \cdot 14 = 2 \cdot k \]
Which simplifies to:
\[ 14 = 2k \]

7. **Solving for \( k \)**:
Dividing both sides by 2:
\[ k = \frac{14}{2} = 7 \]

Thus, the value of \( k \) for which the system is inconsistent is \( k = 7 \).

**Final Answer**: \( k = 7 \)