Find a relaton between a and b, for which the system of equations 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 relationship between \( a \) and \( b \) such that the system of equations \( ax + 2y = 7 \) and \( 3x + by = 16 \) represents parallel lines, we can follow these steps:

### Step 1: Write the equations in slope-intercept form
We need to convert both equations into the form \( y = mx + c \), where \( m \) is the slope.

**For the first equation:**
\[ ax + 2y = 7 \]
Rearranging gives:
\[ 2y = 7 - ax \]
Dividing by 2:
\[ y = \frac{7}{2} - \frac{a}{2}x \]
Here, the slope \( m_1 = -\frac{a}{2} \).

**For the second equation:**
\[ 3x + by = 16 \]
Rearranging gives:
\[ by = 16 - 3x \]
Dividing by \( b \):
\[ y = \frac{16}{b} - \frac{3}{b}x \]
Here, the slope \( m_2 = -\frac{3}{b} \).

### Step 2: Set the slopes equal to each other
Since the lines are parallel, their slopes must be equal:
\[ m_1 = m_2 \]
Thus, we have:
\[ -\frac{a}{2} = -\frac{3}{b} \]

### Step 3: Eliminate the negatives
Multiplying both sides by -1 gives:
\[ \frac{a}{2} = \frac{3}{b} \]

### Step 4: Cross-multiply to find a relationship
Cross-multiplying yields:
\[ ab = 6 \]

### Conclusion
The relationship between \( a \) and \( b \) for which the given system of equations represents parallel lines is:
\[ ab = 6 \]

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