`1/2x - 1/4y = 5 ,ax - 3y = 20`
In the system of linear equations above, a is a constant . If the system has no solutions , what is the value of a ?

Options

A. 6
B. 2
C. `1/3`
D. `-2`

Answer

To solve the problem, we need to determine the value of \( a \) such that the system of linear equations has no solutions. This occurs when the two lines represented by the equations are parallel.

The given equations are:
1. \( \frac{1}{2}x - \frac{1}{4}y = 5 \)
2. \( ax - 3y = 20 \)

### Step 1: Rewrite the first equation in standard form
To find the slope of the first equation, we can rewrite it in the form \( Ax + By + C = 0 \).

Starting from:
\[ \frac{1}{2}x - \frac{1}{4}y = 5 \]

We can multiply through by 4 to eliminate the fractions:
\[ 4 \left(\frac{1}{2}x\right) - 4 \left(\frac{1}{4}y\right) = 4 \cdot 5 \]
This simplifies to:
\[ 2x - y = 20 \]

Rearranging gives us:
\[ 2x - y - 20 = 0 \]

### Step 2: Identify coefficients from the first equation
From the equation \( 2x - y - 20 = 0 \), we identify:
- \( a_1 = 2 \)
- \( b_1 = -1 \)
- \( c_1 = -20 \)

### Step 3: Rewrite the second equation in standard form
The second equation is already in the form \( ax - 3y = 20 \). We can rearrange it:
\[ ax - 3y - 20 = 0 \]

From this equation, we identify:
- \( a_2 = a \)
- \( b_2 = -3 \)
- \( c_2 = -20 \)

### Step 4: Set up the condition for no solutions
For the system to have no solutions, the ratios of the coefficients must be equal, but the ratio of the constants must be different:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{and} \quad \frac{c_1}{c_2} \neq \frac{a_1}{a_2} \]

Substituting in our values:
\[ \frac{2}{a} = \frac{-1}{-3} \]

This simplifies to:
\[ \frac{2}{a} = \frac{1}{3} \]

### Step 5: Cross-multiply to solve for \( a \)
Cross-multiplying gives:
\[ 2 \cdot 3 = 1 \cdot a \]
\[ 6 = a \]

### Step 6: Verify the condition for no solutions
Now we check the condition for the constants:
\[ \frac{c_1}{c_2} = \frac{-20}{-20} = 1 \]
Since \( \frac{2}{6} = \frac{1}{3} \) and \( 1 \neq \frac{1}{3} \), the condition for no solutions is satisfied.

Thus, the value of \( a \) is:
\[ \boxed{6} \]