Through (-3,6) and perpendicular to y=12x-4

To find the equation of the line that passes through the point (-3, 6) and is perpendicular to the line \( y = 12x - 4 \), follow these steps:

### Step 1: Find the slope of the given line
The equation \( y = 12x - 4 \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. In this case, the slope of the given line is \( 12 \).

### Step 2: Find the slope of the perpendicular line
Two lines are **perpendicular** if the product of their slopes is \( -1 \). Let \( m_1 \) be the slope of the given line (which is 12) and \( m_2 \) be the slope of the perpendicular line. Therefore:

\[
m_1 \times m_2 = -1
\]

Substituting \( m_1 = 12 \):

\[
12 \times m_2 = -1
\]

Solving for \( m_2 \):

\[
m_2 = \frac{-1}{12}
\]

Thus, the slope of the perpendicular line is \( \frac{-1}{12} \).

### Step 3: Use point-slope form to find the equation of the perpendicular line
The point-slope form of a line 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 know the slope \( m = \frac{-1}{12} \) and the point \( (-3, 6) \). Substituting these values into the point-slope form:

\[
y - 6 = \frac{-1}{12}(x - (-3))
\]

Simplifying:

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

Distribute \( \frac{-1}{12} \):

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

Add 6 to both sides:

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

Convert 6 to a fraction with a denominator of 4:

\[
y = \frac{-1}{12}x - \frac{1}{4} + \frac{24}{4}
\]

Simplifying:

\[
y = \frac{-1}{12}x + \frac{23}{4}
\]

### Final Answer:
The equation of the line that passes through (-3, 6) and is perpendicular to \( y = 12x - 4 \) is:

\[
y = \frac{-1}{12}x + \frac{23}{4}
\]

Would you like more details on any step? 

Here are some related questions:
1. How do you find the slope of a line from its equation?
2. What are the different forms of a line's equation?
3. How do you find the equation of a line parallel to a given line?
4. What is the geometric meaning of perpendicular lines?
5. How do you graph a line with a fractional slope?

**Tip:** Perpendicular lines have slopes that are negative reciprocals of each other.

Learn how to find the equation of a line that passes through the point (-3, 6) and is perpendicular to y=12x-4. This detailed solution covers key concepts such as perpendicular slopes, slope-intercept form, and point-slope form. Ideal for students in Grades 9-11.

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