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The problem asks for the equation of a line (Second Street) that is **perpendicular** to another line (First Street) and passes through a given point \((4, 7)\). First, let's analyze the equation of First Street and then use the information to find the equation of Second Street.

1. **First Street Equation**:  
   The equation given for First Street is:
   \[
   y = \frac{1}{2}x + 1
   \]
   This is in slope-intercept form \(y = mx + b\), where \(m = \frac{1}{2}\) is the slope of First Street.

2. **Perpendicular Lines**:  
   If two lines are perpendicular, their slopes are negative reciprocals. The negative reciprocal of \(\frac{1}{2}\) is:
   \[
   m_{\text{Second Street}} = -\frac{1}{\frac{1}{2}} = -2
   \]
   Thus, the slope of Second Street is \(-2\).

3. **Finding the y-intercept**:  
   Now, we use the point \((4, 7)\) that lies on Second Street to find the y-intercept. The equation of Second Street is:
   \[
   y = -2x + b
   \]
   Substituting the point \((4, 7)\) into this equation:
   \[
   7 = -2(4) + b
   \]
   \[
   7 = -8 + b
   \]
   \[
   b = 7 + 8 = 15
   \]
   Therefore, the equation of Second Street is:
   \[
   y = -2x + 15
   \]

### Final Answer:
The equation of Second Street is:
\[
y = -2x + 15
\]

Would you like more details on any part of this? Here are some related questions you might want to explore:

1. How do you find the slope of a line from its equation?
2. What does it mean for two lines to be perpendicular in terms of their slopes?
3. How can you verify if a point lies on a given line?
4. How would you find the equation of a line parallel to First Street instead of perpendicular?
5. What is the geometric significance of slope-intercept form in coordinate geometry?

**Tip:** Always remember that perpendicular slopes are negative reciprocals, and parallel lines have identical slopes.

Suppose a coordinate plane were transposed over the map of a city such that First Street had the equation y = 1/2x + 1. If Second Street is perpendicular to First Street and passes through the point (4, 7), what would be the equation of Second Street?

Learn how to find the equation of a line that is perpendicular to y = 1/2x + 1 and passes through the point (4, 7). This problem demonstrates the concept of perpendicular slopes and the slope-intercept form of a line. Suitable for students in Grades 9-11.

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