Question

1. determine which of the lines, if any, are parallel or perpendicular.

line a: ( y = 4x + 6 )
line b: ( y - 3 = -\frac{1}{4}(x - 4) )
line c: ( 8y + 2x = 9 )

1. determine which of the lines, if any, are parallel or perpendicular.

graph with points ((-2, 3)), ((0, 4)), ((4, 4)), ((-2, -1)), ((0, -4)), ((4, 0)) and lines

1. extra credit. grab a piece of graph paper. look at both equations for question 1. write and graph an equation parallel to 1a going through ((2, 6)), and write and graph an equation perpendicular to 1b going through ((0, 2)).

Explanation:

Problem 6

Step 1: Find slopes of each line

• Line a: \( y = 4x + 6 \) is in slope - intercept form \( y=mx + b \), so slope \( m_a=4 \).
• Line b: \( y - 3=-\frac{1}{4}(x - 4) \) is in point - slope form \( y - y_1=m(x - x_1) \), so slope \( m_b =-\frac{1}{4} \).
• Line c: Rewrite \( 8y+2x = 9 \) in slope - intercept form. Solve for \( y \):

\( 8y=-2x + 9\)
\( y=-\frac{2}{8}x+\frac{9}{8}=-\frac{1}{4}x+\frac{9}{8} \), so slope \( m_c=-\frac{1}{4} \).

Step 2: Compare slopes for parallel and perpendicular

• Parallel lines have equal slopes. Since \( m_b=m_c =-\frac{1}{4} \), Line b and Line c are parallel.
• Perpendicular lines have slopes that are negative reciprocals (product is - 1). Check \( m_a\times m_b=4\times(-\frac{1}{4})=-1 \) and \( m_a\times m_c=4\times(-\frac{1}{4})=-1 \). So Line a is perpendicular to both Line b and Line c.

Answer:

Line b and Line c are parallel. Line a is perpendicular to Line b and Line a is perpendicular to Line c.

Problem 7

First, identify the lines by their points:

• Let's assume the lines are:
• Line 1: Passes through \((-2,-1)\) and \((0,4)\)
• Line 2: Passes through \((-2,3)\) and \((4,0)\)
• Line 3: Passes through \((0, - 4)\) and \((4,4)\)
• Line 4: Vertical line (passes through \((0,4)\) and \((0,-4)\)) (undefined slope)

Step 1: Calculate slopes

• Slope of Line 1: \( m_1=\frac{4 - (-1)}{0-(-2)}=\frac{5}{2} \)
• Slope of Line 2: \( m_2=\frac{0 - 3}{4-(-2)}=\frac{-3}{6}=-\frac{1}{2} \)
• Slope of Line 3: \( m_3=\frac{4-(-4)}{4 - 0}=\frac{8}{4}=2 \)
• Line 4: Vertical line, slope is undefined.

Step 2: Compare slopes

• Check for parallel: No two non - vertical lines have the same slope. The vertical line has an undefined slope and no other line is vertical, so no parallel lines among non - vertical lines.
• Check for perpendicular:
• For non - vertical lines, if \( m_1\times m_2=\frac{5}{2}\times(-\frac{1}{2})=-\frac{5}{4}

eq - 1 \)

• \( m_1\times m_3=\frac{5}{2}\times2 = 5

eq - 1 \)

• \( m_2\times m_3=-\frac{1}{2}\times2=-1 \)

So Line 2 (through \((-2,3)\) and \((4,0)\)) and Line 3 (through \((0,-4)\) and \((4,4)\)) are perpendicular.