Find a relation between a and b, for which the system of equation ax + 2y = 7 and 3x + by = 16 represents parallel lines.

Options

A. a -b = 5
B. a + 2b = 7
C. ab = 6
D. `(a)/(2b)`

Answer

To find the relation between \( a \) and \( b \) such that the system of equations \( ax + 2y = 7 \) and \( 3x + by = 16 \) represents parallel lines, we follow these steps:

### Step 1: Rewrite the equations in standard form
We can rewrite the given equations in the standard form \( A_1x + B_1y + C_1 = 0 \) and \( A_2x + B_2y + C_2 = 0 \).

1. The first equation \( ax + 2y = 7 \) can be rewritten as:
\[ ax + 2y - 7 = 0 \]
Here, \( A_1 = a \), \( B_1 = 2 \), and \( C_1 = -7 \).

2. The second equation \( 3x + by = 16 \) can be rewritten as:
\[ 3x + by - 16 = 0 \]
Here, \( A_2 = 3 \), \( B_2 = b \), and \( C_2 = -16 \).

### Step 2: Set up the condition for parallel lines
For two lines to be parallel, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{and} \quad \frac{A_1}{A_2} \neq \frac{C_1}{C_2} \]

### Step 3: Apply the condition
From our equations, we have:
\[ \frac{a}{3} = \frac{2}{b} \]

### Step 4: Cross-multiply to find a relation
Cross-multiplying gives:
\[ a \cdot b = 3 \cdot 2 \]
\[ ab = 6 \]

### Step 5: Conclusion
Thus, the relation between \( a \) and \( b \) for which the lines represented by the equations are parallel is:
\[ ab = 6 \]