−3x + y = 6
ax + 2y = 4
In the system of equations above, a is a constant. For which of the following values of a does the system have no solution?

Options

A. `-6`
B. `-3`
C. 3
D. 6

Explanation

To determine the value of \( a \) for which the system of equations has no solution, we can analyze the given equations:

1. The first equation is:
\[ -3x + y = 6 \]

2. The second equation is:
\[ ax + 2y = 4 \]

### Step 1: Identify the coefficients
From the first equation, we can identify the coefficients:
- \( a_1 = -3 \)
- \( b_1 = 1 \)
- \( c_1 = 6 \)

From the second equation:
- \( a_2 = a \)
- \( b_2 = 2 \)
- \( c_2 = 4 \)

### Step 2: Use the condition for no solution
A system of linear equations has no solution if the following condition holds:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]

Substituting the identified coefficients into this condition gives:
\[ \frac{-3}{a} = \frac{1}{2} \]

### Step 3: Cross-multiply to solve for \( a \)
Cross-multiplying the equation results in:
\[ -3 \cdot 2 = 1 \cdot a \]
This simplifies to:
\[ -6 = a \]

### Conclusion
Thus, the value of \( a \) for which the system of equations has no solution is:
\[ \boxed{-6} \]