Given that the system of equations `3x + ky = 1` and `6x + 3y = 2` . For what value of k does this system have infinite solutions ?

Answer

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

1. \( 3x + ky = 1 \)
2. \( 6x + 3y = 2 \)

has infinite solutions, we need to analyze the conditions under which two linear equations represent the same line. This occurs when the equations are proportional to each other.

### Step-by-step Solution:

**Step 1: Rewrite the second equation.**
We start with the second equation \( 6x + 3y = 2 \). We can simplify this by dividing the entire equation by 2:

\[ \frac{6x}{2} + \frac{3y}{2} = \frac{2}{2} \]

This simplifies to:

\[ 3x + \frac{3}{2}y = 1 \]

**Step 2: Compare the two equations.**
Now we have two equations:

1. \( 3x + ky = 1 \)
2. \( 3x + \frac{3}{2}y = 1 \)

For these two equations to represent the same line, the coefficients of \( y \) must be equal. Thus, we set:

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

**Step 3: Conclusion.**
The value of \( k \) that allows the system to have infinite solutions is:

\[ \boxed{\frac{3}{2}} \]