Question

the equations of three lines are given below.

line 1: (y = 2x - 6)
line 2: (6x - 3y = -9)
line 3: (y = -2x + 1)

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:

🆕 New Concept Discovered: Parallel and Perpendicular Lines
Comparing slopes of linear equations

Step 1: Find the slope of Line 1

Line 1 is given in slope-intercept form, \( y = mx + b \), where \( m \) is the slope:

$$ y = 2x - 6 $$

The slope of Line 1 is:

$$ m_1 = 2 $$

Step 2: Find the slope of Line 2

Line 2 is given in standard form:

$$ 6x - 3y = -9 $$

Convert it to slope-intercept form by solving for \( y \):

$$ -3y = -6x - 9 $$

$$ y = 2x + 3 $$

The slope of Line 2 is:

$$ m_2 = 2 $$

Step 3: Find the slope of Line 3

Line 3 is given in slope-intercept form:

$$ y = -2x + 1 $$

The slope of Line 3 is:

$$ m_3 = -2 $$

Step 4: Compare the slopes

• Parallel lines have equal slopes (\( m_a = m_b \)).
• Perpendicular lines have slopes that are negative reciprocals of each other (\( m_a \cdot m_b = -1 \)).

Let's evaluate each pair:

• Line 1 and Line 2:

$$ m_1 = 2, \quad m_2 = 2 $$

Since the slopes are equal (\( 2 = 2 \)), the lines are Parallel.

• Line 1 and Line 3:

$$ m_1 = 2, \quad m_3 = -2 $$

The slopes are neither equal nor negative reciprocals (\( 2 \cdot -2 = -4
eq -1 \)). The relationship is Neither.

• Line 2 and Line 3:

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

The slopes are neither equal nor negative reciprocals (\( 2 \cdot -2 = -4
eq -1 \)). The relationship is Neither.

Answer:

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