Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. y−6=14(x+2)y−6=14(x+2) y=−4x−10y=−4x−10 2x−y=72x−y=7 3x−4y=−83x−4y=−8 y=−23x+7y=−23x+7 −2x−3y=12

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we need to analyze their slopes based on the equations provided.

1. For the first pair:

   • \( y - 6 = 14(x + 2) \)

     • Rearranging gives: \[ y = 14x + 28 + 6 -> y = 14x + 34 \]
     • The slope of the first line is \( m_1 = 14 \).

   • \( y = -4x - 10 \)

     • The slope of the second line is \( m_2 = -4 \).

   • Since the slopes \( m_1 \) and \( m_2 \) are neither equal (which would indicate parallel lines) nor the product of the slopes equals -1 (which would indicate perpendicular lines), these lines are intersecting.

2. For the second pair:

   • \( 2x - y = 7 \)

     • Rearranging gives: \[ y = 2x - 7 \]
     • The slope of the first line is \( m_1 = 2 \).

   • \( 3x - 4y = -8 \)

     • Rearranging gives: \[ 4y = 3x + 8 \implies y = \frac{3}{4}x + 2 \]
     • The slope of the second line is \( m_2 = \frac{3}{4} \).

   • Since \( m_1 \cdot m_2 = 2 \cdot \frac{3}{4} = \frac{3}{2} \), which is not -1, and the slopes are not equal, these lines are also intersecting.

3. For the third pair:

   • \( y = -\frac{2}{3}x + 7 \)

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

   • \( -2x - 3y = 12 \)

     • Rearranging gives: \[ 3y = -2x - 12 \implies y = -\frac{2}{3}x - 4 \]
     • The slope of the second line is \( m_2 = -\frac{2}{3} \).

   • Since \( m_1 \) is equal to \( m_2 \), the lines are parallel.

In summary:

• The first pair is intersecting.
• The second pair is intersecting.
• The third pair is parallel.