Question

the equations of three lines are given below.
line 1: $6x - 4y = 2$
line 2: $3y = 2x + 7$
line 3: $y = \dfrac{2}{3}x - 3$
for each pair of lines, determine whether they are parallel, perpendicular, or neit
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 \(6x - 4y = 2\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
\(-4y=-6x + 2\)
\(y=\frac{-6}{-4}x-\frac{2}{4}\)
\(y = \frac{3}{2}x-\frac{1}{2}\), so slope of Line 1 (\(m_1\)) is \(\frac{3}{2}\).

Step2: Find slope of Line 2

Rewrite \(3y = 2x + 7\) in slope - intercept form.
\(y=\frac{2}{3}x+\frac{7}{3}\), so slope of Line 2 (\(m_2\)) is \(\frac{2}{3}\).

Step3: Find slope of Line 3

The equation of Line 3 is \(y=\frac{2}{3}x - 3\), so slope of Line 3 (\(m_3\)) is \(\frac{2}{3}\).

Step4: Analyze Line 1 and Line 2

Two lines are parallel if \(m_1=m_2\), perpendicular if \(m_1\times m_2=- 1\).
\(m_1=\frac{3}{2}\), \(m_2=\frac{2}{3}\). \(m_1\times m_2=\frac{3}{2}\times\frac{2}{3}=1
eq - 1\) and \(m_1
eq m_2\). So Line 1 and Line 2 are neither parallel nor perpendicular.

Step5: Analyze Line 1 and Line 3

\(m_1=\frac{3}{2}\), \(m_3=\frac{2}{3}\). \(m_1\times m_3=\frac{3}{2}\times\frac{2}{3}=1
eq - 1\) and \(m_1
eq m_3\). So Line 1 and Line 3 are neither parallel nor perpendicular.

Step6: Analyze Line 2 and Line 3

\(m_2=\frac{2}{3}\), \(m_3=\frac{2}{3}\). Since \(m_2 = m_3\), Line 2 and Line 3 are parallel.

Answer:

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