Question

the equations of three lines are given below.
line 1: $y = 2x + 6$
line 2: $y = -2x - 3$
line 3: $6x - 3y = -3$
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 \(y = m_2x + b_2\), they are parallel if \(m_1 = m_2\), perpendicular if \(m_1\times m_2=- 1\), else neither. First, find slopes of all lines. Line 1: \(y = 2x+6\), so \(m_1 = 2\). Line 2: \(y=-2x - 3\), so \(m_2=-2\). For Line 3: \(6x-3y=-3\), solve for \(y\): \(3y = 6x + 3\), \(y = 2x+1\), so \(m_3 = 2\).

Step2: Analyze Line 1 and Line 2

\(m_1 = 2\), \(m_2=-2\). Check parallel: \(2
eq - 2\), not parallel. Check perpendicular: \(2\times(-2)=-4
eq - 1\), so neither.

Step3: Analyze Line 1 and Line 3

\(m_1 = 2\), \(m_3 = 2\). Since \(m_1=m_3\), they are parallel.

Step4: Analyze Line 2 and Line 3

\(m_2=-2\), \(m_3 = 2\). Check parallel: \(-2
eq2\), not parallel. Check perpendicular: \((-2)\times2=-4
eq - 1\), so neither.

Answer:

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