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Let's break down the problem step by step to find the equation of the parallel street.

1. **Find the slope of the given street:**
   The given street passes through the points \((4, 7)\) and \((3, 3)\). The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the coordinates of the points:
   \[
   m = \frac{3 - 7}{3 - 4} = \frac{-4}{-1} = 4
   \]
   So, the slope of the given street is \(4\).

2. **Equation of the parallel street:**
   Since parallel lines have the same slope, the parallel street will also have a slope of \(4\). The equation of the line is in the form:
   \[
   y = mx + b
   \]
   where \(m = 4\) and \(b\) is the y-intercept, which we need to find.

3. **Find the y-intercept:**
   The parallel street passes through the point \((-2, 4)\). Substitute \((-2, 4)\) and \(m = 4\) into the equation \(y = 4x + b\):
   \[
   4 = 4(-2) + b
   \]
   Simplifying:
   \[
   4 = -8 + b
   \]
   Solving for \(b\):
   \[
   b = 4 + 8 = 12
   \]

4. **Final equation:**
   The equation of the parallel street is:
   \[
   y = 4x + 12
   \]

So, the correct answer is the first option: **\(y = 4x + 12\)**.

Do you want further details or have any questions?

Here are 5 questions to expand on this topic:
1. How do you find the slope of a line given two points?
2. What is the significance of the slope for parallel lines?
3. How do you use a point to find the y-intercept of a line?
4. Can two lines be parallel if their slopes are different?
5. How do you graph a linear equation on a coordinate plane?

**Tip:** When determining whether lines are parallel, always compare their slopes. Parallel lines will always have equal slopes!

A design engineer is mapping out a new neighborhood with parallel streets. If one street passes through (4, 7) and (3, 3), what is the equation for a parallel street that passes through (-2, 4)?

Learn how to determine the equation of a parallel line that passes through a given point, with a step-by-step solution. This problem involves calculating the slope using the slope formula and applying the equation of a line. Suitable for students in Grades 8-10.

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