Value of k for which system of equations `kx + 2y = 5, 3x + y =1 ` has unique solution is

Options

A. `k =1 `
B. `k =2 `
C. `k =3`
D. all are true

Answer

To find the value of \( k \) for which the system of equations

\[ kx + 2y = 5 \]
\[ 3x + y = 1 \]

has a unique solution, we can use the condition that the ratios of the coefficients must not be equal. Specifically, we need to ensure that:

\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

where \( a_1 \) and \( b_1 \) are the coefficients of \( x \) and \( y \) from the first equation, and \( a_2 \) and \( b_2 \) are the coefficients from the second equation.

### Step 1: Identify the coefficients
From the first equation \( kx + 2y = 5 \):
- \( a_1 = k \)
- \( b_1 = 2 \)

From the second equation \( 3x + y = 1 \):
- \( a_2 = 3 \)
- \( b_2 = 1 \)

### Step 2: Set up the ratio condition
We need to ensure that:

\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

Substituting the values we identified:

\[ \frac{k}{3} \neq \frac{2}{1} \]

### Step 3: Simplify the inequality
This simplifies to:

\[ \frac{k}{3} \neq 2 \]

### Step 4: Solve for \( k \)
To find the value of \( k \) that makes the left side equal to the right side, we multiply both sides by 3:

\[ k \neq 6 \]

### Conclusion
Thus, the system of equations has a unique solution for all values of \( k \) except \( k = 6 \).

### Final Answer
The value of \( k \) for which the system of equations has a unique solution is:

\[ k \neq 6 \]

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