For which value of p, the given system of equations has a unique solution?
`x + 2y = 1, x + py = 5`

Options

A. `p=2`
B. `p=0`
C. `p ne 2`
D. `pne 0`

Answer

To determine the value of \( p \) for which the given system of equations has a unique solution, we need to analyze the two equations:

1. \( x + 2y = 1 \) (Equation 1)
2. \( x + py = 5 \) (Equation 2)

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

- For Equation 1:
\[ x + 2y - 1 = 0 \quad \Rightarrow \quad A_1 = 1, B_1 = 2, C_1 = -1 \]

- For Equation 2:
\[ x + py - 5 = 0 \quad \Rightarrow \quad A_2 = 1, B_2 = p, C_2 = -5 \]

### Step 2: Apply the condition for a unique solution
For the system of equations to have a unique solution, the ratio of the coefficients of \( x \) and \( y \) must not be equal. This can be expressed as:

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

Substituting the values we found:

\[ \frac{1}{1} \neq \frac{2}{p} \]

This simplifies to:

\[ 1 \neq \frac{2}{p} \]

### Step 3: Solve the inequality
To solve the inequality \( 1 \neq \frac{2}{p} \), we can rewrite it as:

\[ p \neq 2 \]

### Conclusion
Thus, the system of equations will have a unique solution for all values of \( p \) except \( p = 2 \).

### Final Answer
The value of \( p \) for which the given system of equations has a unique solution is:
\[ p \neq 2 \]