Question

the equations of three lines are given below.
line 1: $3x - 9y = -18$
line 2: $y = -\frac{1}{3}x - 6$
line 3: $y = 3x + 2$
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 \(3x - 9y = -18\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope).
Subtract \(3x\) from both sides: \(-9y=-3x - 18\).
Divide by \(-9\): \(y=\frac{-3}{-9}x+\frac{-18}{-9}\), so \(y = \frac{1}{3}x + 2\). The slope of Line 1 (\(m_1\)) is \(\frac{1}{3}\).

Step2: Identify slopes of Line 2 and Line 3

Line 2: \(y =-\frac{1}{3}x - 6\), slope (\(m_2\)) is \(-\frac{1}{3}\).
Line 3: \(y = 3x+2\), slope (\(m_3\)) is \(3\).

Step3: Analyze Line 1 and Line 2

Compare \(m_1=\frac{1}{3}\) and \(m_2 =-\frac{1}{3}\).

• Parallel lines have equal slopes: \(\frac{1}{3}

eq-\frac{1}{3}\), not parallel.

• Perpendicular lines have slopes that are negative reciprocals (\(m_1\times m_2=- 1\)): \(\frac{1}{3}\times(-\frac{1}{3})=-\frac{1}{9}

eq - 1\), not perpendicular. So Line 1 and Line 2: Neither.

Step4: Analyze Line 1 and Line 3

Compare \(m_1=\frac{1}{3}\) and \(m_3 = 3\).

• Parallel: \(\frac{1}{3}

eq3\), not parallel.

• Perpendicular: \(\frac{1}{3}\times3 = 1

eq - 1\), not perpendicular. So Line 1 and Line 3: Neither. Wait, correction: Wait, \(\frac{1}{3}\) and \(3\): The negative reciprocal of \(\frac{1}{3}\) is \(- 3\), not \(3\). Wait, no: Wait, slope of Line 1 is \(\frac{1}{3}\), slope of Line 3 is \(3\). \(\frac{1}{3}\times3 = 1
eq - 1\), so neither? Wait, no, wait: Wait, the slope of Line 1 is \(\frac{1}{3}\), Line 3 is \(3\). The product is \(1\), not \(- 1\), so neither. Wait, but let's re - check Line 1's slope. Line 1: \(3x-9y=-18\), \(9y = 3x + 18\), \(y=\frac{3}{9}x + 2=\frac{1}{3}x + 2\), correct. Line 3: \(y = 3x+2\), slope \(3\). So \(\frac{1}{3}\) and \(3\): not equal, product is \(1
eq - 1\), so neither.

Step5: Analyze Line 2 and Line 3

Compare \(m_2=-\frac{1}{3}\) and \(m_3 = 3\).

• Parallel: \(-\frac{1}{3}

eq3\), not parallel.

• Perpendicular: \(-\frac{1}{3}\times3=-1\), so they are perpendicular.

Answer:

Line 1 and Line 2: Neither
Line 1 and Line 3: Neither
Line 2 and Line 3: Perpendicular