For the pair of equations `lambdax + 3y + 7 = 0` and ` 2x + 6y - 14 = 0`. To have infinitely many solutions, the value of `lambda` should be 1 . Is the statement true ? Give reasons.

Options

A. yes
B. no
C. can not say anything
D. none of these

Explanation

To determine whether the statement "For the pair of equations \( \lambda x + 3y + 7 = 0 \) and \( 2x + 6y - 14 = 0 \), to have infinitely many solutions, the value of \( \lambda \) should be 1" is true, we need to analyze the conditions for infinitely many solutions in a system of linear equations.

### Step-by-Step Solution:

1. **Write the equations in standard form**:
- Equation 1: \( \lambda x + 3y + 7 = 0 \)
- Equation 2: \( 2x + 6y - 14 = 0 \)

2. **Identify coefficients**:
- For Equation 1:
- \( a_1 = \lambda \)
- \( b_1 = 3 \)
- \( c_1 = 7 \)
- For Equation 2:
- \( a_2 = 2 \)
- \( b_2 = 6 \)
- \( c_2 = -14 \)

3. **Condition for infinitely many solutions**:
- For the system to have infinitely many solutions, the following condition must hold:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

4. **Calculate the ratios**:
- Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{\lambda}{2} \]
- Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \]
- Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{7}{-14} = -\frac{1}{2} \]

5. **Set up the equations**:
- From \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \):
\[ \frac{\lambda}{2} = \frac{1}{2} \]
This gives:
\[ \lambda = 1 \]
- From \( \frac{a_1}{a_2} = \frac{c_1}{c_2} \):
\[ \frac{\lambda}{2} = -\frac{1}{2} \]
This gives:
\[ \lambda = -1 \]

6. **Conclusion**:
- We found two different values for \( \lambda \) that satisfy the conditions for infinitely many solutions: \( \lambda = 1 \) and \( \lambda = -1 \).
- Therefore, the statement "the value of \( \lambda \) should be 1" is **false** because there is no unique value for \( \lambda \) that guarantees infinitely many solutions.

### Final Answer:
The statement is false. There is no unique value for \( \lambda \) that can be chosen to ensure infinitely many solutions.