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We are tasked with writing an equation for the line that is perpendicular to the given line \( y = \frac{1}{2}x + 2 \) and passes through the point \( P(7, -4) \). Let's break it down step by step.

### Step 1: Find the slope of the perpendicular line
The slope of the given line is \( m = \frac{1}{2} \). Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the line perpendicular to this one will be:

\[
m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2
\]

### Step 2: Use the point-slope form of the equation
The point-slope form of a line's equation is:

\[
y - y_1 = m(x - x_1)
\]

Where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. We have \( m = -2 \) and the point \( (x_1, y_1) = (7, -4) \).

Substitute these values into the point-slope form:

\[
y - (-4) = -2(x - 7)
\]

Simplifying:

\[
y + 4 = -2(x - 7)
\]

### Step 3: Final equation in point-slope form
Simplifying the right-hand side:

\[
y + 4 = -2x + 14
\]

Finally, subtract 4 from both sides:

\[
y = -2x + 10
\]

This is the equation of the line perpendicular to \( y = \frac{1}{2}x + 2 \) and passing through the point \( (7, -4) \), written in point-slope form.

Would you like further details or have any questions?

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Here are some follow-up questions:
1. Can you explain why the slope of the perpendicular line is the negative reciprocal?
2. How would you write this equation in slope-intercept form?
3. What would the equation look like if the point were \( (0, 0) \)?
4. How does the process change if you're finding a parallel line instead of a perpendicular one?
5. Can you confirm that \( (7, -4) \) lies on the perpendicular line by substituting it into the equation?

**Tip:** Always remember that the slopes of perpendicular lines are negative reciprocals!

Write an equation of the line perpendicular to the given line that contains P(7, -4); y = (1/2)x + 2. Write an equation for the line in point-slope form.

Learn how to write an equation of a line perpendicular to y = 1/2x + 2 that passes through the point P(7, -4) using the point-slope form. Understand the relationship between slopes of perpendicular lines and key algebraic concepts.

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