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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=−23x+7 −2x−3y=12 2x−y=7 3x−4y=−8 y−6=14(x+2) y=−4x−10

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To determine if the pairs of lines are parallel, perpendicular, or intersecting, we need to examine the slopes of the lines.

First, let's express each equation in slope-intercept form (y = mx + b), where m is the slope.

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

    • Slope (m1) = -2/3

  2. Line 2: Rearranging \( -2x - 3y = 12 \): \[ 3y = -2x - 12 \Rightarrow y = -\frac{2}{3}x - 4 \]

    • Slope (m2) = -2/3

Since m1 = m2, the lines are parallel.

  1. Line 3: Rearranging \( 2x - y = 7 \): \[ y = 2x - 7 \]

    • Slope (m3) = 2

  2. Line 4: Rearranging \( 3x - 4y = -8 \): \[ 4y = 3x + 8 \Rightarrow y = \frac{3}{4}x + 2 \]

    • Slope (m4) = 3/4

Since m3 is not equal to m4 and their product \( 2 \cdot \frac{3}{4} = \frac{3}{2} \neq -1\), the lines are intersecting.

  1. Line 5: Rearranging \( y - 6 = 14(x + 2) \): \[ y = 14x + 28 + 6 = 14x + 34 \]

    • Slope (m5) = 14

  2. Line 6: \( y = -4x - 10 \)

    • Slope (m6) = -4

Since m5 * m6 = 14 * -4 = -56, the lines are intersecting.

Summary:

  1. First Pair: Parallel
  2. Second Pair: Intersecting
  3. Third Pair: Intersecting

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