Question

the equations of three lines are given below.
line 1: $y = -\frac{5}{2}x - 5$
line 2: $2y = -5x + 5$
line 3: $4x - 10y = -6$
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

Line 1: \( y = -\frac{5}{2}x - 5 \), slope \( m_1 = -\frac{5}{2} \)

Step2: Find slope of Line 2

Line 2: \( 2y = -5x + 5 \) → \( y = -\frac{5}{2}x + \frac{5}{2} \), slope \( m_2 = -\frac{5}{2} \)

Step3: Find slope of Line 3

Line 3: \( 4x - 10y = -6 \) → \( -10y = -4x - 6 \) → \( y = \frac{4}{10}x + \frac{6}{10} = \frac{2}{5}x + \frac{3}{5} \), slope \( m_3 = \frac{2}{5} \)

Step4: Compare Line 1 and Line 2

\( m_1 = m_2 = -\frac{5}{2} \), so parallel.

Step5: Compare Line 1 and Line 3

\( m_1 \times m_3 = -\frac{5}{2} \times \frac{2}{5} = -1 \), so perpendicular.

Step6: Compare Line 2 and Line 3

\( m_2 \times m_3 = -\frac{5}{2} \times \frac{2}{5} = -1 \), so perpendicular.

Answer:

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