Question

the equations of three lines are given below. line 1: 6x - 4y = 6 line 2: 2y = 3x + 5 line 3: $y = \frac{3}{2}x - 5$ for each pair of lines, determine whether they are parallel, perpendicular, or neither. line 1 and line 2: parallel perpendicular neither line 1 and line 3: parallel perpendicular neither line 2 and line 3: parallel perpendicular neither

Explanation:

Step1: Find slope of Line 1

Rewrite \(6x - 4y = 6\) in slope - intercept form (\(y=mx + b\), where \(m\) is slope).
\(-4y=-6x + 6\), so \(y=\frac{-6}{-4}x-\frac{6}{4}=\frac{3}{2}x-\frac{3}{2}\). Slope of Line 1 (\(m_1\)) is \(\frac{3}{2}\).

Step2: Find slope of Line 2

Rewrite \(2y = 3x + 5\) in slope - intercept form.
\(y=\frac{3}{2}x+\frac{5}{2}\). Slope of Line 2 (\(m_2\)) is \(\frac{3}{2}\).

Step3: Find slope of Line 3

Line 3 is \(y=\frac{3}{2}x - 5\), so slope of Line 3 (\(m_3\)) is \(\frac{3}{2}\).

Step4: Compare slopes for Line 1 and Line 2

Since \(m_1=m_2=\frac{3}{2}\), lines with equal slopes are parallel. So Line 1 and Line 2 are parallel.

Step5: Compare slopes for Line 1 and Line 3

Since \(m_1 = m_3=\frac{3}{2}\), lines with equal slopes are parallel. So Line 1 and Line 3 are parallel.

Step6: Compare slopes for Line 2 and Line 3

Since \(m_2=m_3=\frac{3}{2}\), lines with equal slopes are parallel. So Line 2 and Line 3 are parallel.

Answer:

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