Question

the equations of three lines are given below.
line 1: $y = -\frac{3}{2}x + 8$
line 2: $6x + 4y = 2$
line 3: $2y = -3x + 7$
for each pair of lines, determine whether they are parallel, perpendicular, or neither.
line 1 and line 2: $\circ$ parallel $\circ$ perpendicular $\circ$ neither
line 1 and line 3: $\circ$ parallel $\circ$ perpendicular $\circ$ neither
line 2 and line 3: $\circ$ parallel $\circ$ perpendicular $\circ$ neither

Explanation:

Step1: Find slopes of all lines

For Line 1: \( y = -\frac{3}{2}x + 8 \), slope \( m_1 = -\frac{3}{2} \).
For Line 2: \( 6x + 4y = 2 \) → \( 4y = -6x + 2 \) → \( y = -\frac{6}{4}x + \frac{2}{4} \) → \( y = -\frac{3}{2}x + \frac{1}{2} \), slope \( m_2 = -\frac{3}{2} \).
For Line 3: \( 2y = -3x + 7 \) → \( y = -\frac{3}{2}x + \frac{7}{2} \), slope \( m_3 = -\frac{3}{2} \).

Step2: Compare slopes for parallel/perpendicular

• Parallel: slopes equal (\( m_1 = m_2 = m_3 \)).
• Perpendicular: product of slopes = -1 (here, \( (-\frac{3}{2})(-\frac{3}{2}) = \frac{9}{4}

eq -1 \), so not perpendicular).

Line 1 and Line 2:

\( m_1 = m_2 = -\frac{3}{2} \), so Parallel.

Line 1 and Line 3:

\( m_1 = m_3 = -\frac{3}{2} \), so Parallel.

Line 2 and Line 3:

\( m_2 = m_3 = -\frac{3}{2} \), so Parallel.

Answer:

Line 1 and Line 2: Parallel
Line 1 and Line 3: Parallel
Line 2 and Line 3: Parallel