Question

the equations of three lines are given below.
line 1: $y = -2x + 7$
line 2: $6x + 3y = 9$
line 3: $y = -2x - 6$
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: Recall slope rules

Parallel lines: equal slopes. Perpendicular lines: slopes multiply to -1. Slope-intercept form: \(y = mx + b\), \(m\) is slope.

Step2: Find slope of Line 1

Line 1: \(y = -2x + 7\), so slope \(m_1 = -2\).

Step3: Find slope of Line 2

Line 2: \(6x + 3y = 9\). Solve for \(y\):
\(3y = -6x + 9\)
\(y = -2x + 3\). Slope \(m_2 = -2\).

Step4: Find slope of Line 3

Line 3: \(y = -2x - 6\), slope \(m_3 = -2\).

Step5: Compare Line 1 and Line 2

\(m_1 = -2\), \(m_2 = -2\). Equal slopes → Parallel.

Step6: Compare Line 1 and Line 3

\(m_1 = -2\), \(m_3 = -2\). Equal slopes → Parallel.

Step7: Compare Line 2 and Line 3

\(m_2 = -2\), \(m_3 = -2\). Equal slopes → Parallel.

Answer:

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