Line p contains the points `(-1, 8) and (9, k)`. If line p is parallel to line q whose equation of `3x+4y=7`. What is the value of k?

Answer

To find the value of \( k \) such that line \( p \) is parallel to line \( q \), we will follow these steps:

### Step 1: Find the slope of line \( p \)
Line \( p \) passes through the points \((-1, 8)\) and \((9, k)\). The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (-1, 8)\) and \((x_2, y_2) = (9, k)\). Therefore, the slope \( m_p \) of line \( p \) is:

\[ m_p = \frac{k - 8}{9 - (-1)} = \frac{k - 8}{10} \]

### Step 2: Find the slope of line \( q \)
The equation of line \( q \) is given as \( 3x + 4y = 7 \). We need to convert this into slope-intercept form \( y = mx + c \).

Rearranging the equation:

\[ 4y = -3x + 7 \]
\[ y = -\frac{3}{4}x + \frac{7}{4} \]

From this, we can see that the slope \( m_q \) of line \( q \) is:

\[ m_q = -\frac{3}{4} \]

### Step 3: Set the slopes equal to each other
Since lines \( p \) and \( q \) are parallel, their slopes must be equal:

\[ m_p = m_q \]

Substituting the slopes we found:

\[ \frac{k - 8}{10} = -\frac{3}{4} \]

### Step 4: Solve for \( k \)
To solve for \( k \), we cross-multiply:

\[ 4(k - 8) = -30 \]

Expanding this gives:

\[ 4k - 32 = -30 \]

Adding 32 to both sides:

\[ 4k = 2 \]

Dividing both sides by 4:

\[ k = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{\frac{1}{2}} \]