Question

for problems 15 & 16, use the part of a city map shown.
15, which street is parallel to 1st ave?
a. 2nd ave
b. main road
c. central ave
d. d street

1. a city planner wants to build a road perpendicular to d street. what is the slope of the new road? show your work!
2. what is the equation of a line that is parallel to the line ( y = 2x + 7 ) and passes through the point ( (-2,4) )? show your work!

Explanation:

Step1: Find the slope of 1st Ave

1st Ave is a vertical line. The slope of a vertical line is undefined.

Step2: Determine the parallel street

Parallel lines have the same slope. 2nd Ave is also a vertical line (slope undefined), Main Road has a slope (not undefined), Central Ave has a slope (not undefined), D Street has a slope (not undefined). So the street parallel to 1st Ave is 2nd Ave.

Step3: Find the slope of D Street

Use two points on D Street. Let's take \((- 4,6)\) and \((6,6)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(y_1 = 6,y_2 = 6,x_1=-4,x_2 = 6\). Then \(m=\frac{6 - 6}{6-(-4)}=\frac{0}{10}=0\).

Step4: Find the slope of the new road (perpendicular to D Street)

If two lines are perpendicular and one has a slope \(m_1\) and the other has a slope \(m_2\), then \(m_1\times m_2=- 1\). Since \(m_1 = 0\) (slope of D Street), the slope of the line perpendicular to a horizontal line (slope \(m = 0\)) is undefined.

Step5: Find the equation of the line \(y = 2x+7\) passing through \((2,4)\)

The slope - intercept form of a line is \(y=mx + b\). The slope of the line parallel to \(y = 2x+7\) is \(m = 2\) (parallel lines have equal slopes). Substitute \(x = 2,y = 4,m = 2\) into \(y=mx + b\) to find \(b\).
\(4=2\times2 + b\), \(4 = 4 + b\), so \(b=0\). The equation is \(y = 2x\).

Answer:

1. A. 2nd Ave
2. The slope of the new road (perpendicular to D Street) is undefined.
3. The equation of the line is \(y = 2x\)