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The problem asks for the equation of a line that is perpendicular to a given line, \( y = \frac{1}{2}x + 4 \), and contains the point \( P(8, -4) \). We are to express the equation in point-slope form.

### Step 1: Identify the slope of the given line
The equation of the given line is in slope-intercept form: 
\[
y = mx + b
\]
where \( m \) is the slope. For the line \( y = \frac{1}{2}x + 4 \), the slope \( m_1 = \frac{1}{2} \).

### Step 2: Find the slope of the perpendicular line
The slopes of perpendicular lines are negative reciprocals of each other. So, the slope of the perpendicular line, \( m_2 \), is:
\[
m_2 = -\frac{1}{\frac{1}{2}} = -2
\]

### Step 3: Use the point-slope form
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. Here, we are given the point \( P(8, -4) \), and the slope is \( -2 \). Substituting these values:
\[
y - (-4) = -2(x - 8)
\]
Simplifying:
\[
y + 4 = -2(x - 8)
\]
\[
y + 4 = -2x + 16
\]
\[
y = -2x + 16 - 4
\]
\[
y = -2x + 12
\]

Thus, the equation of the line in point-slope form is:
\[
y + 4 = -2(x - 8)
\]

Would you like more details or have any further questions?

### Related questions:
1. How do you convert from point-slope form to slope-intercept form?
2. What is the general method for finding the equation of a line given two points?
3. How would you find the equation of a line parallel to \( y = \frac{1}{2}x + 4 \) through the point \( P(8, -4) \)?
4. What is the significance of the negative reciprocal relationship for perpendicular lines?
5. How do you graph lines using point-slope form on the coordinate plane?

### Tip:
When working with perpendicular lines, always remember that their slopes multiply to \( -1 \) (i.e., \( m_1 \times m_2 = -1 \)).

Write an equation of the line perpendicular to the given line that contains P(8, -4). y = (1/2)x + 4. Write the equation in point-slope form.

Learn how to find the equation of a line perpendicular to y = (1/2)x + 4 that passes through the point P(8, -4) using point-slope form. This solution covers algebraic concepts such as finding slopes of perpendicular lines and using the point-slope formula.

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