For which value(s) of `lamda`, do the pair of linear equations `lamda x+ y= lamda^(2) and x+ lamda y= 1` have
a unique solution?

Answer

To find the value(s) of \( \lambda \) for which the pair of linear equations

\[ \lambda x + y = \lambda^2 \]

and

\[ x + \lambda y = 1 \]

has a unique solution, we will follow these steps:

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

1. For the first equation \( \lambda x + y - \lambda^2 = 0 \):
- \( A_1 = \lambda \)
- \( B_1 = 1 \)
- \( C_1 = -\lambda^2 \)

2. For the second equation \( x + \lambda y - 1 = 0 \):
- \( A_2 = 1 \)
- \( B_2 = \lambda \)
- \( C_2 = -1 \)

### Step 2: Use the condition for a unique solution
For the system of equations to have a unique solution, the condition is:

\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]

Substituting the values we identified:

\[ \frac{\lambda}{1} \neq \frac{1}{\lambda} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:

\[ \lambda^2 \neq 1 \]

### Step 4: Solve the inequality
The inequality \( \lambda^2 \neq 1 \) implies:

\[ \lambda \neq 1 \quad \text{and} \quad \lambda \neq -1 \]

### Conclusion
Thus, the values of \( \lambda \) for which the pair of linear equations has a unique solution are:

\[ \lambda \in \mathbb{R} \text{ such that } \lambda \neq 1 \text{ and } \lambda \neq -1 \]