Question

suppose we are given the following.
line 1 passes through \\((-4, -3)\\) and \\((8, 6)\\).
line 2 passes through \\((-1, 0)\\) and \\((0, 2)\\).
line 3 passes through \\((0, -7)\\) and \\((-3, -4)\\).

(a) find the slope of each line.
slope of line 1:
slope of line 2:
slope of line 3:

(b) for each pair of lines, determine whether they are parallel, perpendicular, or neither.
line 1 and line 2: ( ) parallel ( ) perpendicular ( ) neither
line 1 and line 3: ( ) parallel ( ) perpendicular ( ) neither
line 2 and line 3: ( ) parallel ( ) perpendicular ( ) neither

Explanation:

⚡ Using what you learned: slope of a line · parallel and perpendicular lines_equations

Step 1: Slope of Line 1

Line 1 passes through \( (-4, -3) \) and \( (8, 0) \).

$$ m_1 = \frac{0 - (-3)}{8 - (-4)} = \frac{3}{12} = \frac{1}{4} $$

Step 2: Slope of Line 2

Line 2 passes through \( (-1, 0) \) and \( (0, 2) \).

$$ m_2 = \frac{2 - 0}{0 - (-1)} = \frac{2}{1} = 2 $$

Step 3: Slope of Line 3

Line 3 passes through \( (0, -7) \) and \( (-3, -4) \).

$$ m_3 = \frac{-4 - (-7)}{-3 - 0} = \frac{3}{-3} = -1 $$

Step 4: Compare Line 1 and Line 2

$$ m_1 = \frac{1}{4}, \quad m_2 = 2 $$

• Not equal (\( \frac{1}{4}

eq 2 \)) \(
ightarrow\) Not parallel.

• Not negative reciprocals (\( \frac{1}{4} \cdot 2 = \frac{1}{2}

eq -1 \)) \(
ightarrow\) Not perpendicular.

• Relationship: Neither

Step 5: Compare Line 1 and Line 3

$$ m_1 = \frac{1}{4}, \quad m_3 = -1 $$

• Not equal (\( \frac{1}{4}

eq -1 \)) \(
ightarrow\) Not parallel.

• Not negative reciprocals (\( \frac{1}{4} \cdot (-1) = -\frac{1}{4}

eq -1 \)) \(
ightarrow\) Not perpendicular.

• Relationship: Neither

Step 6: Compare Line 2 and Line 3

$$ m_2 = 2, \quad m_3 = -1 $$

• Not equal (\( 2

eq -1 \)) \(
ightarrow\) Not parallel.

• Not negative reciprocals (\( 2 \cdot (-1) = -2

eq -1 \)) \(
ightarrow\) Not perpendicular.

• Relationship: Neither

Answer:

Part (a)

• Slope of Line 1: \( \frac{1}{4} \)
• Slope of Line 2: \( 2 \)
• Slope of Line 3: \( -1 \)

Part (b)

• Line 1 and Line 2: Neither
• Line 1 and Line 3: Neither
• Line 2 and Line 3: Neither