If the straight lines `3x-5y=7and4x+ay+9=0` are perpendicular to one another, find the value of a.

Answer

To find the value of \( a \) such that the lines \( 3x - 5y = 7 \) and \( 4x + ay + 9 = 0 \) are perpendicular, we will follow these steps:

### Step 1: Convert the first line to slope-intercept form
The first line is given by:
\[ 3x - 5y = 7 \]
To convert this to the slope-intercept form \( y = mx + c \), we can rearrange it:
\[ -5y = -3x + 7 \]
Dividing every term by -5 gives:
\[ y = \frac{3}{5}x - \frac{7}{5} \]
From this equation, we can identify the slope \( m_1 \):
\[ m_1 = \frac{3}{5} \]

### Step 2: Convert the second line to slope-intercept form
The second line is given by:
\[ 4x + ay + 9 = 0 \]
Rearranging this equation, we have:
\[ ay = -4x - 9 \]
Dividing every term by \( a \) gives:
\[ y = -\frac{4}{a}x - \frac{9}{a} \]
From this equation, we can identify the slope \( m_2 \):
\[ m_2 = -\frac{4}{a} \]

### Step 3: Use the condition for perpendicular lines
For two lines to be perpendicular, the product of their slopes must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \left(\frac{3}{5}\right) \cdot \left(-\frac{4}{a}\right) = -1 \]
This simplifies to:
\[ -\frac{12}{5a} = -1 \]
Removing the negative sign from both sides gives:
\[ \frac{12}{5a} = 1 \]

### Step 4: Solve for \( a \)
To solve for \( a \), we can cross-multiply:
\[ 12 = 5a \]
Dividing both sides by 5:
\[ a = \frac{12}{5} \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{\frac{12}{5}} \]

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