Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)

Question

y=−2/3x+7
−2x−3y=12

2x−y=7
3x−4y=−8

y−6=1/4(x+2)
y=−4x−10

Answer 1

To determine whether pairs of lines are parallel, perpendicular, or intersecting, we need to analyze their slopes.

1. For the first pair of lines:

   • Line 1: \( y = -\frac{2}{3}x + 7 \)

     • The slope \( m_1 = -\frac{2}{3} \)

   • Line 2: Convert to slope-intercept form: \[ -2x - 3y = 12 \] \[ -3y = 2x + 12 \] \[ y = -\frac{2}{3}x - 4 \]

     • The slope \( m_2 = -\frac{2}{3} \)

   Conclusion: Since \( m_1 = m_2 \), the lines are parallel.

2. For the second pair of lines:

   • Line 1: Convert to slope-intercept form: \[ 2x - y = 7 \] \[ -y = -2x + 7 \] \[ y = 2x - 7 \]

     • The slope \( m_1 = 2 \)

   • Line 2: Convert to slope-intercept form: \[ 3x - 4y = -8 \] \[ -4y = -3x - 8 \] \[ y = \frac{3}{4}x + 2 \]

     • The slope \( m_2 = \frac{3}{4} \)

   Conclusion: Since \( m_1 \neq m_2 \) and \( m_1 \cdot m_2 \neq -1 \) (i.e., \( 2 \cdot \frac{3}{4} = \frac{3}{2} \)), the lines are intersecting.

3. For the third pair of lines:

   • Line 1: \( y - 6 = \frac{1}{4}(x + 2) \)

     • Convert to slope-intercept form: \[ y - 6 = \frac{1}{4}x + \frac{1}{2} \] \[ y = \frac{1}{4}x + \frac{1}{2} + 6 \] \[ y = \frac{1}{4}x + \frac{13}{2} \]
     • The slope \( m_1 = \frac{1}{4} \)

   • Line 2: \( y = -4x - 10 \)

     • The slope \( m_2 = -4 \)

   Conclusion: The slopes \( m_1 \) and \( m_2 \) are not equal (i.e., \( m_1 \cdot m_2 = \frac{1}{4} \cdot -4 = -1 \)), indicating that the lines are perpendicular.

Summary of Results:

1. First pair: Parallel
2. Second pair: Intersecting
3. Third pair: Perpendicular