Find the equation of a line perpendicular toy, equals, 2, x, minus, 6y=2x−6that passes through the point left parenthesis, minus, 8, comma, minus, 2, right parenthesis(−8,−2).

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.

Find the equation of a line perpendicular to y = 2x - 6 that passes through the point (-8, -2).

Learn how to find the equation of a line perpendicular to y = 2x - 6 that passes through the point (-8, -2). This detailed step-by-step solution explains the key concepts of slope-intercept form, point-slope form, and perpendicular line properties, suitable for students in Grades 8-10.

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