To find the line perpendicular to the line $y = -\frac{1}{2}x - 7$, we need to follow these steps:

Step 1: Identify the slope of the given line

The equation $y = -\frac{1}{2}x - 7$ is already in slope-intercept form, $y = mx + b$, where $m$ is the slope. The slope of this line is:

$m_1 = -\frac{1}{2}$

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, if the original slope is $-\frac{1}{2}$, the slope of the perpendicular line is:

$m_2 = -\left(\frac{1}{-1/2}\right) = 2$

Step 3: Check the multiple-choice options

Now, we want to find the equation of a line with a slope of 2. We will check each option for this.

• Option 1: $2x + y = 2$
  This can be rewritten in slope-intercept form: $y = -2x + 2$ The slope is $-2$, so this is not correct.

• Option 2: $x - 2y = 6$
  This can be rewritten in slope-intercept form: $-2y = -x + 6 \quad \Rightarrow \quad y = \frac{1}{2}x - 3$ The slope is $\frac{1}{2}$, so this is not correct.

• Option 3: $y - 2x = 4$
  This can be rewritten as: $y = 2x + 4$ The slope is $2$, which matches the slope of a line perpendicular to the given line. This is the correct answer.

• Option 4: $x + 2y = -8$
  This can be rewritten in slope-intercept form: $2y = -x - 8 \quad \Rightarrow \quad y = -\frac{1}{2}x - 4$ The slope is $-\frac{1}{2}$, so this is not correct.

Final Answer:

The equation that represents a line perpendicular to $y = -\frac{1}{2}x - 7$ is:

$\boxed{y - 2x = 4}$

Do you want more details on how I converted each equation into slope-intercept form or how perpendicular slopes work?

Here are 5 questions to deepen understanding:

1. What is the slope of the line $x - 2y = 6$?
2. How do you find the equation of a line perpendicular to a given line passing through a specific point?
3. How do negative reciprocals relate to perpendicular slopes?
4. Why does changing the slope change the orientation of the line?
5. Can two lines with the same slope ever be perpendicular?

Tip: Remember, two lines are perpendicular if the product of their slopes is $-1$.