Write an equation for the line that passes through the given point and is
perpendicular to the graph of the given equation.
4. y = −2x − 1; (2, −1) 5. y + 4 = −2__
3(x − 2); (4, −2) 6. x − 6y = −2; (−5, 6)

To find the equation of a line that passes through a given point and is perpendicular to a given line, we follow these steps:

1. **Identify the slope of the given line.**
2. **Find the perpendicular slope**, which is the negative reciprocal of the slope of the given line.
3. **Use the point-slope form** of a line equation: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the given point.

Let's solve each problem step by step:

### 4. \( y = -2x - 1 \); Point: \( (2, -1) \)

- The slope of the given line \( y = -2x - 1 \) is \( m = -2 \).
- The slope of the perpendicular line is the negative reciprocal of \( -2 \), which is \( \frac{1}{2} \).
- Using the point-slope form:
  \[
  y - (-1) = \frac{1}{2}(x - 2)
  \]
  Simplify:
  \[
  y + 1 = \frac{1}{2}(x - 2)
  \]
  Distribute the \( \frac{1}{2} \):
  \[
  y + 1 = \frac{1}{2}x - 1
  \]
  Subtract 1 from both sides:
  \[
  y = \frac{1}{2}x - 2
  \]
- **Equation of the perpendicular line**: \( y = \frac{1}{2}x - 2 \).

### 5. \( y + 4 = -\frac{2}{3}(x - 2) \); Point: \( (4, -2) \)

- First, identify the slope of the given line: The equation is already in point-slope form. The slope is \( m = -\frac{2}{3} \).
- The slope of the perpendicular line is the negative reciprocal of \( -\frac{2}{3} \), which is \( \frac{3}{2} \).
- Using the point-slope form:
  \[
  y - (-2) = \frac{3}{2}(x - 4)
  \]
  Simplify:
  \[
  y + 2 = \frac{3}{2}(x - 4)
  \]
  Distribute the \( \frac{3}{2} \):
  \[
  y + 2 = \frac{3}{2}x - 6
  \]
  Subtract 2 from both sides:
  \[
  y = \frac{3}{2}x - 8
  \]
- **Equation of the perpendicular line**: \( y = \frac{3}{2}x - 8 \).

### 6. \( x - 6y = -2 \); Point: \( (-5, 6) \)

- First, rewrite the given equation in slope-intercept form \( y = mx + b \):
  \[
  x - 6y = -2
  \]
  Solve for \( y \):
  \[
  -6y = -x - 2
  \]
  Divide by -6:
  \[
  y = \frac{1}{6}x + \frac{1}{3}
  \]
- The slope of the given line is \( m = \frac{1}{6} \).
- The slope of the perpendicular line is the negative reciprocal of \( \frac{1}{6} \), which is \( -6 \).
- Using the point-slope form:
  \[
  y - 6 = -6(x - (-5))
  \]
  Simplify:
  \[
  y - 6 = -6(x + 5)
  \]
  Distribute the \( -6 \):
  \[
  y - 6 = -6x - 30
  \]
  Add 6 to both sides:
  \[
  y = -6x - 24
  \]
- **Equation of the perpendicular line**: \( y = -6x - 24 \).

---

If you have any questions or need further details, feel free to ask! 

Here are 5 related questions:
1. What is the formula for finding the slope of a line from its equation?
2. How do you derive the negative reciprocal of a given slope?
3. What is the geometric interpretation of perpendicular lines on a graph?
4. Can two perpendicular lines have the same slope?
5. How do you convert a line equation from standard form to slope-intercept form?

**Tip:** The negative reciprocal of a slope \( m \) is \( -\frac{1}{m} \). Always use this to find the perpendicular slope!

Write an equation for the line that passes through the given point and is perpendicular to the graph of the given equation. 4. y = −2x − 1; (2, −1) 5. y + 4 = −2/3(x − 2); (4, −2) 6. x − 6y = −2; (−5, 6)

Learn how to write the equation of a line passing through a given point and perpendicular to a given line. This solution involves key algebraic concepts such as slopes, perpendicular lines, and point-slope form. Suitable for students in Grades 9-10.

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