The value of `lambda` for the pair of equations to have infinitely many solutions is
`lambda x + 3y = -4`
`x - 6y = 8`

Answer

To find the value of `lambda` for the pair of equations to have infinitely many solutions, we need to set up the equations properly and use the condition for infinitely many solutions.

The given equations are:
1. \( \lambda x + 3y = -4 \) (Equation 1)
2. \( x - 6y = 8 \) (Equation 2)

For the two equations to have infinitely many solutions, they must represent the same line. This occurs when the ratios of the coefficients of \(x\), \(y\), and the constant terms are equal. Mathematically, this can be expressed as:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Where:
- \( a_1 = \lambda \), \( b_1 = 3 \), \( c_1 = -4 \) (from Equation 1)
- \( a_2 = 1 \), \( b_2 = -6 \), \( c_2 = 8 \) (from Equation 2)

Now, we can set up the ratios:

1. From the coefficients of \(x\):
\[ \frac{\lambda}{1} = \frac{\lambda}{1} \]

2. From the coefficients of \(y\):
\[ \frac{3}{-6} = -\frac{1}{2} \]

3. From the constant terms:
\[ \frac{-4}{8} = -\frac{1}{2} \]

Now we equate the ratios:

\[ \frac{\lambda}{1} = -\frac{1}{2} \]

To find the value of \( \lambda \), we can solve this equation:

\[ \lambda = -\frac{1}{2} \]

Thus, the value of \( \lambda \) for the pair of equations to have infinitely many solutions is:

\[ \lambda = -\frac{1}{2} \]