We are tasked with finding the equation of a line that is perpendicular to the line $y = 2x - 6$ and passes through the point $(-8, -2)$.

Step 1: Find the slope of the given line

The equation of the given line is in the slope-intercept form $y = mx + b$, where $m$ is the slope. For the given equation:

$y = 2x - 6$

Here, the slope $m = 2$.

Step 2: Find the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. If the slope of the given line is $2$, the slope of the perpendicular line will be:

$m_{\text{perpendicular}} = -\frac{1}{2}$

Step 3: Use the point-slope form of a line

The point-slope form 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. We know the slope of the perpendicular line is $-\frac{1}{2}$, and it passes through the point $(-8, -2)$.

Substitute $m = -\frac{1}{2}$, $x_1 = -8$, and $y_1 = -2$ into the equation:

$y - (-2) = -\frac{1}{2}(x - (-8))$

Simplify:

$y + 2 = -\frac{1}{2}(x + 8)$

Step 4: Simplify the equation

Distribute the slope on the right-hand side:

$y + 2 = -\frac{1}{2}x - 4$

Subtract 2 from both sides to isolate $y$:

$y = -\frac{1}{2}x - 4 - 2$

$y = -\frac{1}{2}x - 6$

Final Answer:

The equation of the line perpendicular to $y = 2x - 6$ that passes through the point $(-8, -2)$ is:

$y = -\frac{1}{2}x - 6$

Would you like more details or have any questions?

Here are 5 related questions you can explore:

1. How do you find the equation of a line parallel to a given line?
2. What is the general form of the equation of a line?
3. How do you convert the slope-intercept form to standard form?
4. How do you calculate the distance between a point and a line?
5. What is the relationship between slopes of perpendicular lines?

Tip: When dealing with perpendicular lines, always remember their slopes are negative reciprocals of each other.