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### #2: 
Write a linear equation in **point-slope form** of the line that passes through the point \( (2, -4) \) and is perpendicular to \( y = \frac{2}{5}x - 4 \).

- **Step 1:** Find the slope of the given line. The slope of \( y = \frac{2}{5}x - 4 \) is \( m = \frac{2}{5} \).
- **Step 2:** For perpendicular lines, the slopes are negative reciprocals. The slope of the perpendicular line is \( m_{\perp} = -\frac{5}{2} \).
- **Step 3:** Use the point-slope formula \( y - y_1 = m(x - x_1) \) with the point \( (2, -4) \) and slope \( -\frac{5}{2} \).
  \[
  y + 4 = -\frac{5}{2}(x - 2)
  \]

### #3: 
Write a linear equation in **point-slope form** of the line that passes through the point \( (-2, 3) \) and is parallel to \( 2x + y = -5 \).

- **Step 1:** Rewrite the given equation in slope-intercept form.
  \[
  2x + y = -5 \implies y = -2x - 5
  \]
  The slope is \( m = -2 \).
- **Step 2:** For parallel lines, the slopes are the same. So the slope of the new line is \( m = -2 \).
- **Step 3:** Use the point-slope form with the point \( (-2, 3) \) and slope \( -2 \).
  \[
  y - 3 = -2(x + 2)
  \]

### #4:
Write a linear equation in **standard form** of the line that passes through the point \( (2, 0) \) and is perpendicular to \( y = \frac{1}{2}x \).

- **Step 1:** The slope of the given line is \( m = \frac{1}{2} \).
- **Step 2:** The slope of the perpendicular line is \( m_{\perp} = -2 \).
- **Step 3:** Use point-slope form to find the equation.
  \[
  y - 0 = -2(x - 2) \implies y = -2x + 4
  \]
- **Step 4:** Convert to standard form.
  \[
  2x + y = 4
  \]

### #5:
Write a linear equation in **point-slope form** of the line that passes through the point \( (-2, -1) \) and is perpendicular to \( 2x - y = -2 \).

- **Step 1:** Rewrite the given equation in slope-intercept form.
  \[
  2x - y = -2 \implies y = 2x + 2
  \]
  The slope is \( m = 2 \).
- **Step 2:** The slope of the perpendicular line is \( m_{\perp} = -\frac{1}{2} \).
- **Step 3:** Use the point-slope form with the point \( (-2, -1) \) and slope \( -\frac{1}{2} \).
  \[
  y + 1 = -\frac{1}{2}(x + 2)
  \]

### #6:
Write a linear equation in **slope-intercept form** of the line that passes through the point \( (3, 0) \) and is perpendicular to \( y = 4 \).

- **Step 1:** The line \( y = 4 \) is a horizontal line with slope \( m = 0 \).
- **Step 

Write a linear equation in point-slope form and standard form for various points and conditions such as perpendicularity and parallelism.

Learn how to write linear equations in point-slope and standard forms, focusing on lines that pass through specific points and are parallel or perpendicular to given lines. Key concepts include point-slope form, slope-intercept form, and the relationship between the slopes of parallel and perpendicular lines. Suitable for Grades 9-11.

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