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We are tasked with finding the equation of a line that:

1. Passes through the point \( (2, -2) \), and
2. Is perpendicular to the line \( y = \frac{2}{5}x + 2 \).

### Step 1: Find the slope of the perpendicular line.
- The given line has the equation \( y = \frac{2}{5}x + 2 \). Here, the slope \( m \) is \( \frac{2}{5} \).
- Since the line we want is perpendicular to this line, its slope \( m_{\perp} \) is the negative reciprocal of \( \frac{2}{5} \). This gives us:
  \[
  m_{\perp} = -\frac{5}{2}
  \]

### Step 2: Use the point-slope form of a line equation.
- The point-slope form is:
  \[
  y - y_1 = m(x - x_1)
  \]
  where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.
- Using the point \( (2, -2) \) and the slope \( -\frac{5}{2} \), the equation becomes:
  \[
  y - (-2) = -\frac{5}{2}(x - 2)
  \]
  Simplifying:
  \[
  y + 2 = -\frac{5}{2}(x - 2)
  \]
  \[
  y + 2 = -\frac{5}{2}x + 5
  \]
  \[
  y = -\frac{5}{2}x + 3
  \]

### Step 3: Convert to standard form.
- Standard form is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.
- To remove the fraction, multiply the entire equation by 2:
  \[
  2y = -5x + 6
  \]
  Now rearrange it into standard form:
  \[
  5x + 2y = 6
  \]

Thus, the equation of the line in standard form is:
\[
\boxed{5x + 2y = 6}
\]

This corresponds to answer choice **D**.

Do you want more details or have any questions?

Here are 5 related questions to further explore:
1. What is the equation of a line parallel to \( y = \frac{2}{5}x + 2 \) through the point \( (2, -2) \)?
2. How do you derive the slope of perpendicular lines from the original slope?
3. What is the significance of point-slope form versus slope-intercept form?
4. How do you convert an equation from slope-intercept form to standard form?
5. How would the process change if you were given a different point?

**Tip:** When finding perpendicular slopes, always remember to take the negative reciprocal of the original slope!

What is the equation in standard form of the line that passes through the point (2, -2) and is perpendicular to the line represented by y = (2/5)x + 2?

Learn how to find the equation in standard form of a line passing through the point (2, -2) and perpendicular to the line y = (2/5)x + 2. The solution includes finding the negative reciprocal slope and converting the equation to standard form. Suitable for students in Grades 9-11.

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