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To solve this problem, we need to find the equation of the line that:

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

### Step 1: Identify the slope of the perpendicular line
The slope of the given line \(y = \frac{2}{5}x + 2\) is \(\frac{2}{5}\). Since the two lines are perpendicular, their slopes must be negative reciprocals of each other.

The negative reciprocal of \(\frac{2}{5}\) is \(-\frac{5}{2}\). Therefore, the slope of the line we are looking for is \(-\frac{5}{2}\).

### Step 2: Use point-slope form to find the equation
The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. Plugging in the point \((2, -2)\) and the slope \(-\frac{5}{2}\), we get:

\[
y - (-2) = -\frac{5}{2}(x - 2)
\]

Simplify:

\[
y + 2 = -\frac{5}{2}(x - 2)
\]

Distribute the slope:

\[
y + 2 = -\frac{5}{2}x + 5
\]

Now, subtract 2 from both sides:

\[
y = -\frac{5}{2}x + 3
\]

### Step 3: Convert to standard form
To convert to standard form \(Ax + By = C\), we eliminate the fraction by multiplying the entire equation by 2:

\[
2y = -5x + 6
\]

Now, move the terms involving \(x\) and \(y\) to one side:

\[
5x + 2y = 6
\]

Thus, the equation of the line in standard form is \(5x + 2y = 6\), which corresponds to option D.

### Final Answer:
**The correct answer is D: \(5x + 2y = 6\).**

Do you want further details or explanations on any part? Here are some related questions to explore:

1. How do you find the slope of a line from its equation in slope-intercept form?
2. What are negative reciprocals, and why do perpendicular lines have slopes that are negative reciprocals?
3. How can you use point-slope form to write the equation of a line given a point and a slope?
4. How do you convert an equation from slope-intercept form to standard form?
5. Why is it necessary to multiply the equation by a factor to eliminate fractions in standard form?

**Tip:** Always double-check the slope for perpendicular lines by taking the negative reciprocal to avoid sign mistakes!

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 of a line in standard form that passes through the point (2, -2) and is perpendicular to the line y = (2/5)x + 2. This solution covers concepts such as point-slope form, standard form of a line, and perpendicular slopes. Suitable for high school students in Grades 9-11.

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