line j is y+6=6(x-1) line k is perpendicular to j and passes through (8,-4) what’s the equation of line k

To find the equation of line \( k \), which is perpendicular to line \( j \) and passes through the point \( (8, -4) \), we can follow these steps:

### Step 1: Rewrite the equation of line \( j \) in slope-intercept form
The given equation of line \( j \) is:

\[
y + 6 = 6(x - 1)
\]

First, solve for \( y \) to put it in slope-intercept form (\( y = mx + b \)):

\[
y + 6 = 6x - 6 \quad \text{(distribute the 6 on the right side)}
\]

\[
y = 6x - 12 \quad \text{(subtract 6 from both sides)}
\]

So, the slope of line \( j \) is \( m_j = 6 \).

### Step 2: Find the slope of line \( k \)
Since line \( k \) is perpendicular to line \( j \), the slope of line \( k \), denoted by \( m_k \), is the negative reciprocal of \( m_j \). Therefore:

\[
m_k = -\frac{1}{6}
\]

### Step 3: Use the point-slope form of the equation of a line
Now that we know the slope of line \( k \) is \( -\frac{1}{6} \) and it passes through the point \( (8, -4) \), we can use the point-slope form of a line equation:

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

Substitute \( m_k = -\frac{1}{6} \), \( x_1 = 8 \), and \( y_1 = -4 \):

\[
y - (-4) = -\frac{1}{6}(x - 8)
\]

\[
y + 4 = -\frac{1}{6}(x - 8)
\]

### Step 4: Simplify the equation
Distribute the slope on the right-hand side:

\[
y + 4 = -\frac{1}{6}x + \frac{8}{6}
\]

Simplify \( \frac{8}{6} \) to \( \frac{4}{3} \):

\[
y + 4 = -\frac{1}{6}x + \frac{4}{3}
\]

Now subtract 4 from both sides:

\[
y = -\frac{1}{6}x + \frac{4}{3} - 4
\]

Convert 4 to a fraction with denominator 3:

\[
y = -\frac{1}{6}x + \frac{4}{3} - \frac{12}{3}
\]

\[
y = -\frac{1}{6}x - \frac{8}{3}
\]

### Final Answer:
The equation of line \( k \) is:

\[
y = -\frac{1}{6}x - \frac{8}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you find the slope of a line given two points?
2. What does it mean geometrically for two lines to be perpendicular?
3. How can you convert between point-slope form and slope-intercept form?
4. What are the steps to find the equation of a parallel line through a given point?
5. How can you determine if two lines are parallel or perpendicular by comparing their slopes?

**Tip:** Always double-check signs when dealing with perpendicular slopes, as they must be negative reciprocals.

Learn how to find the equation of a line perpendicular to y + 6 = 6(x - 1) and passing through the point (8, -4). This problem covers essential algebra and geometry concepts, including slope-intercept form, point-slope form, and perpendicular slopes. Suitable for students in Grades 9-11.

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