The pair of linear equations, kx + 4y = 5 and 3x + 2y = 5 are consistent when,

Options

A. `k ne 6`
B. `k=6`
C. `k ne 3`
D. `k=3`

Answer

To determine when the pair of linear equations \( kx + 4y = 5 \) and \( 3x + 2y = 5 \) are consistent, we need to analyze the conditions for consistency in linear equations.

### Step-by-Step Solution:

1. **Identify the Coefficients**:
For the equations:
- \( kx + 4y = 5 \) can be written in the standard form as \( A_1x + B_1y = C_1 \) where \( A_1 = k \), \( B_1 = 4 \), and \( C_1 = 5 \).
- \( 3x + 2y = 5 \) can be written as \( A_2x + B_2y = C_2 \) where \( A_2 = 3 \), \( B_2 = 2 \), and \( C_2 = 5 \).

2. **Set Up the Consistency Conditions**:
The equations are consistent if they have either a unique solution or infinitely many solutions.
- For infinitely many solutions, the condition is:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]
- For a unique solution, the condition is:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \neq \frac{C_1}{C_2} \]

3. **Calculate the Ratios**:
- Calculate \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{4}{2} = 2 \]
- Calculate \( \frac{C_1}{C_2} \):
\[ \frac{C_1}{C_2} = \frac{5}{5} = 1 \]

4. **Set Up the Conditions for Consistency**:
- For the equations to be consistent, we need:
\[ \frac{k}{3} \neq 2 \quad \text{and} \quad \frac{k}{3} \neq 1 \]

5. **Solve the Inequalities**:
- From \( \frac{k}{3} \neq 2 \):
\[ k \neq 6 \]
- From \( \frac{k}{3} \neq 1 \):
\[ k \neq 3 \]

6. **Conclusion**:
The pair of linear equations \( kx + 4y = 5 \) and \( 3x + 2y = 5 \) are consistent when:
\[ k \neq 6 \quad \text{and} \quad k \neq 3 \]