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Let's break down this problem step by step.

We are given:
- The private plane and the jetliner fly over the point \((-2, 2)\).
- The equation of the jetliner’s path is \( y = \frac{1}{2}x + 3 \).
- The private plane flies perpendicular to the jetliner.

### Step 1: Find the slope of the jetliner’s path
The equation for the jetliner’s path is given as \( y = \frac{1}{2}x + 3 \), which is already in slope-intercept form \( y = mx + b \), where:
- \( m = \frac{1}{2} \) is the slope of the jetliner’s path.

### Step 2: Find the slope of the private plane’s path
Since the private plane flies perpendicular to the jetliner, the slopes of the two lines must be negative reciprocals. If the slope of the jetliner’s path is \( \frac{1}{2} \), then the slope of the private plane’s path is:
\[
m_{\text{private plane}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]

### Step 3: Find the equation of the private plane’s path
We know the slope of the private plane’s path is \( -2 \), and it passes through the point \((-2, 2)\). To find the equation of the line, we use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = -2 \), \( x_1 = -2 \), and \( y_1 = 2 \):
\[
y - 2 = -2(x + 2)
\]
Now, distribute the slope on the right-hand side:
\[
y - 2 = -2x - 4
\]
Finally, solve for \( y \) to put the equation in slope-intercept form:
\[
y = -2x - 2
\]

Thus, the equation of the private plane’s flight path in slope-intercept form is:
\[
y = -2x - 2
\]

Would you like more details, or do you have any questions?

### Here are 5 related questions to deepen understanding:
1. How do you calculate the slope of a line when given two points?
2. What is the relationship between slopes of perpendicular lines?
3. How can you check if two lines are parallel or perpendicular?
4. How can the point-slope form of a line be derived from its slope and a point on the line?
5. How does the y-intercept of a line change if its slope remains constant but the line passes through a different point?

### Tip:
When two lines are perpendicular, their slopes multiply to give \(-1\). This property can help you quickly determine the slope of a perpendicular line.

A private plane and a jetliner both fly over a park that is at (-2, 2) on a map. The path of the jetliner follows the line y = 1/2x + 3. The private plane flies perpendicular to the jetliner. What is the equation in slope-intercept form of the line that represents the private plane's flight path?

Learn how to find the equation of a line in slope-intercept form that is perpendicular to another line and passes through a given point (-2, 2). This problem covers key concepts in linear equations and perpendicular slopes. Suitable for Grades 8-10.

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