determine the equation of the line which is perpendicular to y=3x-2 and passes through (6, 2)

To determine the equation of the line that is perpendicular to \( y = 3x - 2 \) and passes through the point \( (6, 2) \), follow these steps:

### Steps:

1. **Identify the slope of the given line**:
   - The equation of the given line is \( y = 3x - 2 \).
   - The slope \( m_1 \) of this line is \( 3 \).

2. **Determine the slope of the perpendicular line**:
   - The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
   - So, the slope \( m_2 \) of the perpendicular line is:
     \[
     m_2 = -\frac{1}{3}
     \]

3. **Use the point-slope form to find the equation of the perpendicular line**:
   - The point-slope form of a line's equation is:
     \[
     y - y_1 = m(x - x_1)
     \]
   - Here, \( (x_1, y_1) = (6, 2) \) and \( m = -\frac{1}{3} \).
   - Substituting these values into the equation:
     \[
     y - 2 = -\frac{1}{3}(x - 6)
     \]

4. **Simplify the equation**:
   - Distribute the slope on the right side:
     \[
     y - 2 = -\frac{1}{3}x + 2
     \]
   - Add 2 to both sides to get the equation in slope-intercept form:
     \[
     y = -\frac{1}{3}x + 4
     \]

### Result:
The equation of the line that is perpendicular to \( y = 3x - 2 \) and passes through the point \( (6, 2) \) is:
\[
y = -\frac{1}{3}x + 4
\]

Would you like more details or have any questions?

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Here are 5 related questions for further practice:
1. Find the equation of the line perpendicular to \( y = -\frac{1}{2}x + 5 \) that passes through the point \( (4, -3) \).
2. Determine the equation of the line parallel to \( y = 3x - 2 \) that passes through the point \( (1, 4) \).
3. What is the slope of a line perpendicular to \( y = \frac{2}{3}x + 1 \)?
4. Find the point of intersection between the lines \( y = -\frac{1}{3}x + 4 \) and \( y = 3x - 2 \).
5. Convert the equation \( y = -\frac{1}{3}x + 4 \) into standard form \( Ax + By = C \).

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

Learn how to determine the equation of a line perpendicular to y=3x-2 and passing through the point (6, 2). This problem involves understanding slope relationships and applying the point-slope form of a line equation. Suitable for students in Grades 7-9.

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