The graph of 3x-4y=5 is perpendicular to the graph of 2x+ky-10=0. Find k.

Answer

To find the value of \( k \) such that the graph of the equation \( 3x - 4y = 5 \) is perpendicular to the graph of the equation \( 2x + ky - 10 = 0 \), we can follow these steps:

### Step 1: Convert the first equation to slope-intercept form
The first equation is given as:
\[ 3x - 4y = 5 \]
We can rearrange this equation to solve for \( y \):
\[ -4y = -3x + 5 \]
Dividing through by -4 gives:
\[ y = \frac{3}{4}x - \frac{5}{4} \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = \frac{3}{4} \]

### Step 2: Convert the second equation to slope-intercept form
The second equation is given as:
\[ 2x + ky - 10 = 0 \]
Rearranging this equation to solve for \( y \):
\[ ky = -2x + 10 \]
Dividing through by \( k \) gives:
\[ y = -\frac{2}{k}x + \frac{10}{k} \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = -\frac{2}{k} \]

### 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 slopes we found:
\[ \left(\frac{3}{4}\right) \left(-\frac{2}{k}\right) = -1 \]
This simplifies to:
\[ -\frac{6}{4k} = -1 \]
Removing the negative signs gives:
\[ \frac{6}{4k} = 1 \]

### Step 4: Solve for \( k \)
Cross-multiplying gives:
\[ 6 = 4k \]
Dividing both sides by 4:
\[ k = \frac{6}{4} = \frac{3}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{3}{2}} \]