Question

the equations of three lines are given below.
line 1: $y = -3x + 6$
line 2: $y = -3x - 7$
line 3: $6x + 2y = 4$
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: Recall slope properties

For two lines \(y = m_1x + b_1\) and \(y = m_2x + b_2\), parallel if \(m_1 = m_2\), perpendicular if \(m_1 \cdot m_2=-1\). For a line \(Ax + By = C\), slope \(m = -\frac{A}{B}\).

Step2: Find slopes of Line 1, Line 2, Line 3

• Line 1: \(y=-3x + 6\), slope \(m_1=-3\).
• Line 2: \(y=-3x - 7\), slope \(m_2=-3\).
• Line 3: \(6x + 2y = 4\), rewrite as \(2y=-6x + 4\), \(y=-3x + 2\), slope \(m_3=-3\).

Step3: Analyze Line 1 and Line 2

\(m_1 = m_2=-3\), so they are parallel.

Step4: Analyze Line 1 and Line 3

\(m_1 = m_3=-3\), so they are parallel.

Step5: Analyze Line 2 and Line 3

\(m_2 = m_3=-3\), so they are parallel.

Answer:

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