`3/4x-1/2y=12`
`ax-by=9`
The system of equations above has no solutions. If a and b are constants, what is the value of `a/b` ?

Explanation

To solve the problem, we need to determine the value of \( \frac{a}{b} \) for the system of equations given that there are no solutions. This implies that the two lines represented by the equations are parallel.

### Step 1: Rewrite the first equation in slope-intercept form (y = mx + c)
The first equation is:
\[ \frac{3}{4}x - \frac{1}{2}y = 12 \]
To isolate \( y \), we can rearrange the equation:
\[ -\frac{1}{2}y = -\frac{3}{4}x + 12 \]
Multiplying through by -2 to solve for \( y \):
\[ y = \frac{3}{2}x - 24 \]

### Step 2: Identify the slope of the first equation
From the equation \( y = \frac{3}{2}x - 24 \), we can see that the slope \( m_1 \) is:
\[ m_1 = \frac{3}{2} \]

### Step 3: Rewrite the second equation in slope-intercept form
The second equation is:
\[ ax - by = 9 \]
Rearranging this equation to solve for \( y \):
\[ -by = -ax + 9 \]
Dividing through by -b:
\[ y = \frac{a}{b}x - \frac{9}{b} \]

### Step 4: Identify the slope of the second equation
From the equation \( y = \frac{a}{b}x - \frac{9}{b} \), we can see that the slope \( m_2 \) is:
\[ m_2 = \frac{a}{b} \]

### Step 5: Set the slopes equal to each other
Since the two lines are parallel (as there are no solutions), their slopes must be equal:
\[ m_1 = m_2 \]
Thus, we have:
\[ \frac{3}{2} = \frac{a}{b} \]

### Step 6: Solve for \( \frac{a}{b} \)
From the equation above, we can directly conclude:
\[ \frac{a}{b} = \frac{3}{2} \]

### Final Answer
The value of \( \frac{a}{b} \) is:
\[ \frac{3}{2} \]