Question

the equations of three lines are given below.
line 1: $2x - 8y = -8$
line 2: $y = -4x - 7$
line 3: $y = -4x + 4$
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: Find slope of Line 1

Rewrite \(2x - 8y = -8\) in slope - intercept form (\(y = mx + b\), where \(m\) is the slope).
Subtract \(2x\) from both sides: \(-8y=-2x - 8\).
Divide by \(-8\): \(y=\frac{-2}{-8}x+\frac{-8}{-8}\), so \(y = \frac{1}{4}x + 1\). The slope of Line 1 (\(m_1\)) is \(\frac{1}{4}\).

Step2: Identify slopes of Line 2 and Line 3

Line 2: \(y=-4x - 7\), slope (\(m_2\)) is \(-4\).
Line 3: \(y=-4x + 4\), slope (\(m_3\)) is \(-4\).

Step3: Analyze Line 1 and Line 2

Check if parallel: \(m_1=\frac{1}{4}\), \(m_2 = - 4\), \(\frac{1}{4}
eq - 4\), not parallel.
Check if perpendicular: \(m_1\times m_2=\frac{1}{4}\times(-4)=-1\), so they are perpendicular.

Step4: Analyze Line 1 and Line 3

Check if parallel: \(m_1=\frac{1}{4}\), \(m_3=-4\), \(\frac{1}{4}
eq - 4\), not parallel.
Check if perpendicular: \(m_1\times m_3=\frac{1}{4}\times(-4)=-1\), so they are perpendicular.

Step5: Analyze Line 2 and Line 3

Check if parallel: \(m_2=-4\), \(m_3 = - 4\), so \(m_2=m_3\), they are parallel.

Answer:

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